Integrand size = 29, antiderivative size = 616 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7}{3} b c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
7/12*b^2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+1/3*b^2*c^2*d^2*(-c*x+1)*(c*x+1)*( -c^2*d*x^2+d)^(1/2)/x+23/12*b^2*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/ (c*x-1)^(1/2)/(c*x+1)^(1/2)-5/2*b*c^5*d^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*ar ccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)+7/3*b*c^3*d^2*(c*x-1)^(1/2)*(c*x+1 )^(1/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-1/3*b*c*d^2*(c*x-1)^(3/2)* (c*x+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x^2+5/2*c^4*d^2*x*(- c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2+7/3*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*( a+b*arccosh(c*x))^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/3*c^2*d*(-c^2*d*x^2+d)^( 3/2)*(a+b*arccosh(c*x))^2/x-1/3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/ x^3-5/6*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^3/b/(c*x-1)^(1/2)/ (c*x+1)^(1/2)-14/3*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))*ln(1+ (c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/3*b^2*c ^3*d^2*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2 )/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 2.61 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {-8 a b c d^3 x+8 a b c^2 d^3 x^2-8 a^2 d^3 \sqrt {\frac {-1+c x}{1+c x}}+64 a^2 c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}}+8 b^2 c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}}-44 a^2 c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}}-8 b^2 c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}}-12 a^2 c^6 d^3 x^6 \sqrt {\frac {-1+c x}{1+c x}}+20 b^2 c^3 d^3 x^3 (-1+c x) \text {arccosh}(c x)^3-60 a^2 c^3 d^{5/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-6 a b c^3 d^3 x^3 \cosh (2 \text {arccosh}(c x))+6 a b c^4 d^3 x^4 \cosh (2 \text {arccosh}(c x))-112 a b c^3 d^3 x^3 \log (c x)+112 a b c^4 d^3 x^4 \log (c x)-56 b^2 c^3 d^3 x^3 (-1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+3 b^2 c^3 d^3 x^3 \sinh (2 \text {arccosh}(c x))-3 b^2 c^4 d^3 x^4 \sinh (2 \text {arccosh}(c x))+2 b d^3 (-1+c x) \text {arccosh}(c x) \left (4 b c x+8 a \sqrt {\frac {-1+c x}{1+c x}}+8 a c x \sqrt {\frac {-1+c x}{1+c x}}-56 a c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}-56 a c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}+3 b c^3 x^3 \cosh (2 \text {arccosh}(c x))+56 b c^3 x^3 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-6 a c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )-2 b d^3 (-1+c x) \text {arccosh}(c x)^2 \left (-30 a c^3 x^3+4 b \left (-\sqrt {\frac {-1+c x}{1+c x}}-c x \sqrt {\frac {-1+c x}{1+c x}}+7 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+7 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+3 b c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )}{24 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \] Input:
Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]
Output:
(-8*a*b*c*d^3*x + 8*a*b*c^2*d^3*x^2 - 8*a^2*d^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 64*a^2*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 8*b^2*c^2*d^3*x^2*Sqrt[ (-1 + c*x)/(1 + c*x)] - 44*a^2*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 8* b^2*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 12*a^2*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] + 20*b^2*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^3 - 60*a^2*c ^3*d^(5/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x* Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 6*a*b*c^3*d^3*x^3*Cosh[2* ArcCosh[c*x]] + 6*a*b*c^4*d^3*x^4*Cosh[2*ArcCosh[c*x]] - 112*a*b*c^3*d^3*x ^3*Log[c*x] + 112*a*b*c^4*d^3*x^4*Log[c*x] - 56*b^2*c^3*d^3*x^3*(-1 + c*x) *PolyLog[2, -E^(-2*ArcCosh[c*x])] + 3*b^2*c^3*d^3*x^3*Sinh[2*ArcCosh[c*x]] - 3*b^2*c^4*d^3*x^4*Sinh[2*ArcCosh[c*x]] + 2*b*d^3*(-1 + c*x)*ArcCosh[c*x ]*(4*b*c*x + 8*a*Sqrt[(-1 + c*x)/(1 + c*x)] + 8*a*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 56*a*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 56*a*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 3*b*c^3*x^3*Cosh[2*ArcCosh[c*x]] + 56*b*c^3*x^3*Log[1 + E^(-2*ArcCosh[c*x])] - 6*a*c^3*x^3*Sinh[2*ArcCosh[c*x]]) - 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]^2*(-30*a*c^3*x^3 + 4*b*(-Sqrt[(-1 + c*x)/(1 + c*x)] - c* x*Sqrt[(-1 + c*x)/(1 + c*x)] + 7*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 7*c^ 3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)])) + 3*b*c^3*x^3*Sinh[2*ArcCosh[c*x] ]))/(24*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6335 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c \int \frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x^2}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c \left (\int 3 c^2 \sqrt {c x-1} \sqrt {c x+1}dx-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c \left (3 c^2 \int \sqrt {c x-1} \sqrt {c x+1}dx-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \int \sqrt {c x-1} \sqrt {c x+1}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (-\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^2}dx+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6343 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \int \sqrt {c x-1} \sqrt {c x+1}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
Input:
Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]
Output:
$Aborted
Time = 0.63 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a^{2} c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a^{2} c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a^{2} c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a^{2} c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a^{2} c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}+6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-3 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+10 \operatorname {arccosh}\left (c x \right )^{3} x^{3} c^{3}-28 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-28 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}+56 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+28 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+4 c^{3} x^{3}+4 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-56 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(652\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a^{2} c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a^{2} c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a^{2} c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a^{2} c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a^{2} c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}+6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-3 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+10 \operatorname {arccosh}\left (c x \right )^{3} x^{3} c^{3}-28 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-28 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}+56 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+28 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+4 c^{3} x^{3}+4 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-56 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{12 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(652\) |
Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4 /3*a^2*c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a^2*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2 *a^2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a^2*c^4*d^3/(c^2*d)^(1/2)*arctan(( c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/12*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x- 1)^(1/2)/(c*x+1)^(1/2)/x^3*(-6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2* x^4*c^4+6*arccosh(c*x)*c^5*x^5-3*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+10*ar ccosh(c*x)^3*x^3*c^3-28*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2 -28*arccosh(c*x)^2*c^3*x^3+56*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2))^2)*x^3*c^3-3*c^3*x^3*arccosh(c*x)+28*polylog(2,-(c*x+(c*x-1)^(1/2) *(c*x+1)^(1/2))^2)*x^3*c^3-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+4*c^3*x^3 +4*arccosh(c*x)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x*arccosh(c*x))*d^2-1/12 *a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3*(-12*(c*x-1)^( 1/2)*(c*x+1)^(1/2)*arccosh(c*x)*x^4*c^4+6*c^5*x^5+30*arccosh(c*x)^2*c^3*x^ 3-56*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2-56*c^3*x^3*arccosh(c *x)+56*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3-3*c^3*x^3+8*arcco sh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*d^2
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/x^4,x, algorithm="fric as")
Output:
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^4, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Timed out} \] Input:
integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2/x**4,x)
Output:
Timed out
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/x^4,x, algorithm="maxi ma")
Output:
1/6*(10*(-c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d *x^2 + d)^(7/2)/(d*x^3))*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^4 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*log( c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/x^4,x, algorithm="giac ")
Output:
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \] Input:
int(((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^4,x)
Output:
int(((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2))/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a^{2} c^{3} x^{3}+3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+14 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}+12 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}-24 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{2}}d x \right ) a b \,c^{2} x^{3}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-12 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}+12 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) a b \,c^{4} x^{3}+6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c^{4} x^{3}\right )}{6 x^{3}} \] Input:
int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))^2/x^4,x)
Output:
(sqrt(d)*d**2*(15*asin(c*x)*a**2*c**3*x**3 + 3*sqrt( - c**2*x**2 + 1)*a**2 *c**4*x**4 + 14*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt( - c**2*x** 2 + 1)*a**2 + 12*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**4,x)*a*b*x**3 - 24*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**2,x)*a*b*c**2*x**3 + 6*int ((sqrt( - c**2*x**2 + 1)*acosh(c*x)**2)/x**4,x)*b**2*x**3 - 12*int((sqrt( - c**2*x**2 + 1)*acosh(c*x)**2)/x**2,x)*b**2*c**2*x**3 + 12*int(sqrt( - c* *2*x**2 + 1)*acosh(c*x),x)*a*b*c**4*x**3 + 6*int(sqrt( - c**2*x**2 + 1)*ac osh(c*x)**2,x)*b**2*c**4*x**3))/(6*x**3)