Integrand size = 29, antiderivative size = 336 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^2*(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2/c^4/d^2/(- c^2*d*x^2+d)^(1/2)-2/3*(a+b*arccosh(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1 /3*b*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d^2/(-c^2*x^2+1) /(-c^2*d*x^2+d)^(1/2)-10/3*b*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-5/ 3*b^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1 /2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+5/3*b^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 3.68 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {4 a^2 \left (-2+3 c^2 x^2\right )-a b \left (\text {arccosh}(c x) (4-12 \cosh (2 \text {arccosh}(c x)))-2 \sinh (2 \text {arccosh}(c x))+5 \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right ) \left (3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-\sinh (3 \text {arccosh}(c x))\right )\right )-b^2 \left (2+2 \text {arccosh}(c x)^2-2 \left (1+3 \text {arccosh}(c x)^2\right ) \cosh (2 \text {arccosh}(c x))-15 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \left (\log \left (1-e^{-\text {arccosh}(c x)}\right )-\log \left (1+e^{-\text {arccosh}(c x)}\right )\right )+20 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-20 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))+5 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-5 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))\right )}{12 c^4 d \left (d-c^2 d x^2\right )^{3/2}} \]
(4*a^2*(-2 + 3*c^2*x^2) - a*b*(ArcCosh[c*x]*(4 - 12*Cosh[2*ArcCosh[c*x]]) - 2*Sinh[2*ArcCosh[c*x]] + 5*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh [c*x]/2]])*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - Sinh[3*ArcCosh[c*x]]) ) - b^2*(2 + 2*ArcCosh[c*x]^2 - 2*(1 + 3*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c* x]] - 15*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*(Log[1 - E^(-Ar cCosh[c*x])] - Log[1 + E^(-ArcCosh[c*x])]) + 20*((-1 + c*x)/(1 + c*x))^(3/ 2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 20*((-1 + c*x)/(1 + c*x))^ (3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] - 2*ArcCosh[c*x]*Sinh[2*Ar cCosh[c*x]] + 5*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x ]] - 5*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]]))/(12* c^4*d*(d - c^2*d*x^2)^(3/2))
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.98, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {6349, 6327, 6329, 25, 6304, 6318, 3042, 26, 4670, 2715, 2838, 6349, 83, 6318, 3042, 26, 4670, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6349 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{2 c^2}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c^3}+\frac {x (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
(x^2*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*b*Sqrt[- 1 + c*x]*Sqrt[1 + c*x]*(-1/2*b/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(2*c^2*(1 - c^2*x^2)) + ((I/2)*((2*I)*(a + b*ArcCosh[c* x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLo g[2, E^ArcCosh[c*x]]))/c^3))/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*((a + b*Ar cCosh[c*x])^2/(c^2*d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[ 2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/(c^2*d*Sqrt[d - c^2 *d*x^2])))/(3*c^2*d)
3.3.16.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 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] && LtQ[p, -1] && IGtQ[m, 1]
Time = 1.00 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.89
| method | result | size |
| default | \(a^{2} \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-2 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -4 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )\) | \(635\) |
| parts | \(a^{2} \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-2 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -4 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )\) | \(635\) |
a^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(-c^2*d*x^2+d)^(3/2))+b^2*(1 /3*(-d*(c^2*x^2-1))^(1/2)*(3*arccosh(c*x)^2*x^2*c^2+(c*x+1)^(1/2)*arccosh( c*x)*(c*x-1)^(1/2)*c*x+c^2*x^2-2*arccosh(c*x)^2-1)/(c^2*x^2-1)^2/d^3/c^4+5 /3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2*x^2-1)* arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/3*(-d*(c^2*x^2-1))^(1 /2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1) ^(1/2)*(c*x+1)^(1/2))-5/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)/d^3/c^4/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))- 5/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2*x^2-1) *polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+2*a*b*(1/6*(-d*(c^2*x^2-1))^( 1/2)*(6*c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-4*arccosh(c*x ))/(c^2*x^2-1)^2/d^3/c^4-5/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/d^3/c^4/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)+5/6*(-d*(c ^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2*x^2-1)*ln(1+c*x+ (c*x-1)^(1/2)*(c*x+1)^(1/2)))
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(b^2*x^3*arccosh(c*x)^2 + 2*a*b*x^3*arccosh(c*x) + a^2*x^3)*sqrt (-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/6*a*b*c*(2*sqrt(-d)*x/(c^6*d^3*x^2 - c^4*d^3) + 5*sqrt(-d)*log(c*x + 1)/ (c^5*d^3) - 5*sqrt(-d)*log(c*x - 1)/(c^5*d^3)) + 2/3*a*b*(3*x^2/((-c^2*d*x ^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))*arccosh(c*x) + 1/ 3*a^2*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^ 4*d)) + b^2*integrate(x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d *x^2 + d)^(5/2), x)
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]