Integrand size = 29, antiderivative size = 638 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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+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+5/2*c^4*d^2*x*(a+b*arc cosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+23/12*b^2*c^3*d^2*arccosh(c*x)*(-c^2*d*x ^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/2*b*c^5*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)-7/3*b*c^3*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 /3*b*c*d^2*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x -1)^(1/2)/(c*x+1)^(1/2)-7/3*c^3*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)-5/6*c^3*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)-14/3*b*c^3*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)+7/3*b^2*c^3*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 = 2.73 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\frac {-8 a b c d^3 x+8 a b c^2 d^3 x^2-8 a^2 d^3 \sqrt {\frac {-1+c x}{1+c x}}+64 a^2 c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}}+8 b^2 c^2 d^3 x^2 \sqrt {\frac {-1+c x}{1+c x}}-44 a^2 c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}}-8 b^2 c^4 d^3 x^4 \sqrt {\frac {-1+c x}{1+c x}}-12 a^2 c^6 d^3 x^6 \sqrt {\frac {-1+c x}{1+c x}}+20 b^2 c^3 d^3 x^3 (-1+c x) \text {arccosh}(c x)^3-60 a^2 c^3 d^{5/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-6 a b c^3 d^3 x^3 \cosh (2 \text {arccosh}(c x))+6 a b c^4 d^3 x^4 \cosh (2 \text {arccosh}(c x))-112 a b c^3 d^3 x^3 \log (c x)+112 a b c^4 d^3 x^4 \log (c x)-56 b^2 c^3 d^3 x^3 (-1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+3 b^2 c^3 d^3 x^3 \sinh (2 \text {arccosh}(c x))-3 b^2 c^4 d^3 x^4 \sinh (2 \text {arccosh}(c x))+2 b d^3 (-1+c x) \text {arccosh}(c x) \left (4 b c x+8 a \sqrt {\frac {-1+c x}{1+c x}}+8 a c x \sqrt {\frac {-1+c x}{1+c x}}-56 a c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}-56 a c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}+3 b c^3 x^3 \cosh (2 \text {arccosh}(c x))+56 b c^3 x^3 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-6 a c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )-2 b d^3 (-1+c x) \text {arccosh}(c x)^2 \left (-30 a c^3 x^3+4 b \left (-\sqrt {\frac {-1+c x}{1+c x}}-c x \sqrt {\frac {-1+c x}{1+c x}}+7 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+7 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+3 b c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )}{24 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]
(-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 |
| \(\displaystyle -\frac {5}{3} c^2 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x}-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )\right )+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-2 c^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {\left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{3 x^3}\) |
3.2.93.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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_.)*((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_.))*((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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcCosh[c *x])/(f*(m + 1))), x] + (-Simp[b*c*((-d)^p/(f*(m + 1))) Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], 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]), x], x]) / ; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[( m + 1)/2, 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 = 1.29 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.02
| 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 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+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} x^{3} c^{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 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+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 ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-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 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+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} x^{3} c^{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 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+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 ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-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}}\) | \(652\) |
-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*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+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*x^3*c^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*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^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)*c^4*x^4+6*c^5*x^5+30*arccosh(c*x)^2*x^3*c^ 3+56*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3-56*(c*x+1)^(1/2)*ar ccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-56*c^3*x^3*arccosh(c*x)-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
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \]