Integrand size = 29, antiderivative size = 607 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
-5/4*c^2*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2-(-c^2*d*x^2+d)^(5/2 )*(a+b*arccosh(c*x))^2/x-31/64*b^2*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-1/32*b^2 *c^2*d^2*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)-15/8*c^2*d^2*x*(a+b*arcco sh(c*x))^2*(-c^2*d*x^2+d)^(1/2)-89/64*b^2*c*d^2*arccosh(c*x)*(-c^2*d*x^2+d )^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+15/8*b*c^3*d^2*x^2*(a+b*arccosh(c*x))* (-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*d^2*(-c^2*x^2+1)*(a+b *arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/8*b*c*d^ 2*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c* x+1)^(1/2)+c*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/( c*x+1)^(1/2)+5/8*c*d^2*(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/(c*x-1) ^(1/2)/(c*x+1)^(1/2)+2*b*c*d^2*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b^2*c *d^2*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2 )/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 5.26 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\frac {d^2 \left (96 a^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-8-9 c^2 x^2+2 c^4 x^4\right )+1440 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-768 a b \sqrt {d-c^2 d x^2} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x)^2+2 \log (c x)\right )\right )-256 b^2 \sqrt {d-c^2 d x^2} \left (\text {arccosh}(c x) \left (3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x) (3+\text {arccosh}(c x))+6 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )\right )+3 c x \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+384 a b c x \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+64 b^2 c x \sqrt {d-c^2 d x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )-12 a b c x \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-b^2 c x \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )\right )}{768 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(d^2*(96*a^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4) + 1440*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)] *(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 76 8*a*b*Sqrt[d - c^2*d*x^2]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[ c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 256*b^2*Sqrt[d - c^2*d*x^2]*(A rcCosh[c*x]*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(Ar cCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])) + 3*c*x*P olyLog[2, -E^(-2*ArcCosh[c*x])]) + 384*a*b*c*x*Sqrt[d - c^2*d*x^2]*(Cosh[2 *ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 6 4*b^2*c*x*Sqrt[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*Ar cCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]) - 12*a*b*c*x* Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c *x]*Sinh[4*ArcCosh[c*x]]) - b^2*c*x*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4* ArcCosh[c*x]])))/(768*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Result contains complex when optimal does not.
Time = 4.33 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.09, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.966, Rules used = {6343, 6312, 25, 6310, 6298, 101, 43, 6308, 6327, 6329, 40, 40, 43, 6334, 40, 40, 43, 6334, 40, 43, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (\frac {b c d \sqrt {d-c^2 d x^2} \int -x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \int (c x-1)^{3/2} (c x+1)^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-5 c^2 d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c \int (c x-1)^{3/2} (c x+1)^{3/2}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \int \sqrt {c x-1} \sqrt {c x+1}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {1}{4} b c \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-5 c^2 d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{4 c^2}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{4} d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
-(((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x) - 5*c^2*d*((x*(d - c^2 *d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/4 + (3*d*((x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(6*b *c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*Sqrt[d - c^2*d*x^2]*((x^2*(a + b*A rcCosh[c*x]))/2 - (b*c*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh [c*x]/(2*c^3)))/2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/4 - (b*c*d*Sqrt[d - c ^2*d*x^2]*(-1/4*((1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/c^2 + (b*((x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - A rcCosh[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (2*b*c *d^2*Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 + ((1 - c ^2*x^2)^2*(a + b*ArcCosh[c*x]))/4 + (b*c*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) /2 - ArcCosh[c*x]/(2*c)))/2 - (b*c*((x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c*x]/(2*c)))/4))/4 + ( I*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])*L og[1 + E^(-2*ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4)))/ b))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.91.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* (a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1)) Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ b*c + a*d, 0] && IGtQ[m + 1/2, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[x*(a + b*ArcCosh[c*x])^ (n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p )] Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcCosh[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcCosh[c*x])/x), x], x] - Simp[b*c*((-d )^p/(2*p)) Int[(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x]) /; FreeQ [{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G tQ[p, 0] && LtQ[m, -1]
Time = 1.24 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a^{2} c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a^{2} c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-72 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+72 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-33 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+40 \operatorname {arccosh}\left (c x \right )^{3} x c -64 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-64 \operatorname {arccosh}\left (c x \right )^{2} x c +128 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \,\operatorname {arccosh}\left (c x \right )+64 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-8 c^{5} x^{5}-144 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} x c -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(589\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a^{2} c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a^{2} c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-72 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+72 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-33 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+40 \operatorname {arccosh}\left (c x \right )^{3} x c -64 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-64 \operatorname {arccosh}\left (c x \right )^{2} x c +128 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \,\operatorname {arccosh}\left (c x \right )+64 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-8 c^{5} x^{5}-144 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} x c -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(589\) |
-a^2/d/x*(-c^2*d*x^2+d)^(7/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(5/2)-5/4*a^2*c^2*d *x*(-c^2*d*x^2+d)^(3/2)-15/8*a^2*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-15/8*a^2*c ^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/64*b^2 *(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x*(16*(c*x+1)^(1/2)*(c *x-1)^(1/2)*arccosh(c*x)^2*x^4*c^4-8*arccosh(c*x)*c^5*x^5+2*(c*x+1)^(1/2)* (c*x-1)^(1/2)*c^4*x^4-72*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^ 2+72*c^3*x^3*arccosh(c*x)-33*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+40*arccos h(c*x)^3*x*c-64*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2-64*arccosh(c*x) ^2*x*c+128*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-33*c *x*arccosh(c*x)+64*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c)*d^ 2+1/64*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x*(32*(c*x-1 )^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4-8*c^5*x^5-144*(c*x+1)^(1/2)*arc cosh(c*x)*(c*x-1)^(1/2)*c^2*x^2+72*c^3*x^3+120*arccosh(c*x)^2*x*c-128*arcc osh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-128*c*x*arccosh(c*x)+128*ln(1+(c*x+(c *x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-33*c*x)*d^2
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arccosh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x^2, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
-1/8*(10*(-c^2*d*x^2 + d)^(3/2)*c^2*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^2*d^2* x + 15*c*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)/x)*a^2 + integrate ((-c^2*d*x^2 + d)^(5/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^2 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^2, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,d x \]