Integrand size = 22, antiderivative size = 413 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \]
1/3*x*arccosh(a*x)^3/c/(-a^2*c*x^2+c)^(3/2)-x*arccosh(a*x)/c^2/(-a^2*c*x^2 +c)^(1/2)+2/3*x*arccosh(a*x)^3/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*arccosh(a*x)^2 *(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)+2/3*a rccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arc cosh(a*x)^2*ln(1-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1 )^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*ln(-a^2*x^2+1)*(a*x-1)^(1/2)*(a*x+1 )^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arccosh(a*x)*polylog(2,(a*x+(a*x-1)^( 1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1 /2)+polylog(3,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^( 1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-i \pi ^3-\frac {12 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)}{-1+a x}+\frac {6 \text {arccosh}(a x)^2}{1-a^2 x^2}+8 \text {arccosh}(a x)^3+\frac {8 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3}{-1+a x}-\frac {4 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \text {arccosh}(a x)^3}{(-1+a x)^3}-24 \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )+12 \log (a x)+12 \log \left (\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{a x}\right )-24 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )\right )}{12 a c^2 \sqrt {c-a^2 c x^2}} \]
(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((-I)*Pi^3 - (12*a*x*Sqrt[(-1 + a*x) /(1 + a*x)]*ArcCosh[a*x])/(-1 + a*x) + (6*ArcCosh[a*x]^2)/(1 - a^2*x^2) + 8*ArcCosh[a*x]^3 + (8*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/(-1 + a*x) - (4*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - 24*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])] + 12*Log[a*x] + 12*Log[(Sqr t[(-1 + a*x)/(1 + a*x)]*(1 + a*x))/(a*x)] - 24*ArcCosh[a*x]*PolyLog[2, E^( 2*ArcCosh[a*x])] + 12*PolyLog[3, E^(2*ArcCosh[a*x])]))/(12*a*c^2*Sqrt[c - a^2*c*x^2])
Result contains complex when optimal does not.
Time = 2.12 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.71, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6316, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 3011, 2720, 6329, 6315, 240, 7143}
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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{(1-a x)^2 (a x+1)^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{(1-a x)^2 (a x+1)^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{1-a^2 x^2}dx}{c \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \int \frac {a x \text {arccosh}(a x)^2}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \int -i \text {arccosh}(a x)^2 \tan \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \int \text {arccosh}(a x)^2 \tan \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2}{1-e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^2}{1-e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\int \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{2} \int \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\int \frac {\text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{a}+\frac {\text {arccosh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6315 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {-a \int \frac {x}{1-a^2 x^2}dx-\frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}}{a}+\frac {\text {arccosh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}+\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}+\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )+\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\) |
(x*ArcCosh[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(ArcCosh[a*x]^2/(2*a^2*(1 - a^2*x^2)) + (-((x*ArcCosh[a*x])/(Sqrt[- 1 + a*x]*Sqrt[1 + a*x])) + Log[1 - a^2*x^2]/(2*a))/a))/(c^2*Sqrt[c - a^2*c *x^2]) + (2*((x*ArcCosh[a*x]^3)/(c*Sqrt[c - a^2*c*x^2]) + ((3*I)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*((-1/3*I)*ArcCosh[a*x]^3 - (2*I)*(-1/2*(ArcCosh[a*x]^2 *Log[1 - E^(2*ArcCosh[a*x])]) - (ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x] )])/2 + PolyLog[3, E^(2*ArcCosh[a*x])]/4)))/(a*c*Sqrt[c - a^2*c*x^2])))/(3 *c)
3.3.51.3.1 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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp [b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), 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_.)/(((d1_) + (e1_.)*(x_))^(3/2)* ((d2_) + (e2_.)*(x_))^(3/2)), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/( d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])), x] + Simp[b*c*(n/(d1*d2))*Simp[Sqr t[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[x*(( a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
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), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Coth[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_.)*(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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(400)=800\).
Time = 1.32 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.31
| method | result | size |
| default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-3 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \operatorname {arccosh}\left (a x \right ) \left (6 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )+6 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{4} x^{4}+6 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}-9 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}\, a x -18 a^{2} x^{2}-8 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+12\right )}{6 \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) c^{3} a}-\frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (\sqrt {a x -1}\, \sqrt {a x +1}+a x -1\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{3}}{3 c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}\) | \(955\) |
-1/6*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-3*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+ 1)^(1/2)+2*(a*x-1)^(1/2)*(a*x+1)^(1/2))*arccosh(a*x)*(6*a^3*x^3*arccosh(a* x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a^4*x^4*arccosh(a*x)+6*a^3*x^3*(a*x-1)^(1 /2)*(a*x+1)^(1/2)+6*a^4*x^4+6*a^2*x^2*arccosh(a*x)^2-9*a*x*arccosh(a*x)*(a *x-1)^(1/2)*(a*x+1)^(1/2)-12*a^2*x^2*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1)^ (1/2)*a*x-18*a^2*x^2-8*arccosh(a*x)^2+6*arccosh(a*x)+12)/(3*a^6*x^6-10*a^4 *x^4+11*a^2*x^2-4)/c^3/a-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2 )/c^3/a/(a^2*x^2-1)*ln((a*x-1)^(1/2)*(a*x+1)^(1/2)+a*x-1)-(a*x+1)^(1/2)*(a *x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*ln(1+a*x+(a*x-1)^(1/2 )*(a*x+1)^(1/2))+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/ a/(a^2*x^2-1)*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-4/3*(a*x+1)^(1/2)*(a*x-1 )^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^3+2*(a*x+1)^ (1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^ 2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c* (a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^( 1/2)*(a*x+1)^(1/2))-4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c ^3/a/(a^2*x^2-1)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+2*(a*x+1)^(1/ 2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^2*l n(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^ 2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1...
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]