\(\int \frac {\text {arccosh}(a x)^3}{(c-a^2 c x^2)^{5/2}} \, dx\) [69]

Optimal result

Integrand size = 22, antiderivative size = 401 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \] Output:

-x*arccosh(a*x)/c^2/(-a^2*c*x^2+c)^(1/2)-1/2*arccosh(a*x)^2/a/c^2/(a*x-1)^ 
(1/2)/(a*x+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)+1/3*x*arccosh(a*x)^3/c/(-a^2*c*x^ 
2+c)^(3/2)+2/3*x*arccosh(a*x)^3/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*(a*x-1)^(1/2) 
*(a*x+1)^(1/2)*arccosh(a*x)^3/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*(a*x-1)^(1/2)*( 
a*x+1)^(1/2)*arccosh(a*x)^2*ln(1-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/a/c^ 
2/(-a^2*c*x^2+c)^(1/2)+1/2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*ln(-a^2*x^2+1)/a/c^ 
2/(-a^2*c*x^2+c)^(1/2)-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)*polylog( 
2,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+(a*x-1)^ 
(1/2)*(a*x+1)^(1/2)*polylog(3,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/a/c^2/( 
-a^2*c*x^2+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.64 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-i \pi ^3-\frac {12 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)}{-1+a x}+\frac {6 \text {arccosh}(a x)^2}{1-a^2 x^2}+8 \text {arccosh}(a x)^3+\frac {8 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3}{-1+a x}-\frac {4 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \text {arccosh}(a x)^3}{(-1+a x)^3}-24 \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )+12 \log \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)\right )-24 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )\right )}{12 a c^2 \sqrt {c-a^2 c x^2}} \] Input:

Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]
 

Output:

(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[Sqrt[(-1 + a*x)/(1 
 + a*x)]*(1 + 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])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.12 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.73, 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

\(\displaystyle \frac {a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}+\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}}{a}\right )}{c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {3 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )+\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{3} i \text {arccosh}(a x)^3\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}\)

Input:

Int[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 2620

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]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

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]
 

rule 6314

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]
 

rule 6315

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]
 

rule 6316

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]
 

rule 6327

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]
 

rule 6328

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]
 

rule 6329

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]
 

rule 7143

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(388)=776\).

Time = 0.45 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.38

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 \operatorname {arccosh}\left (a x \right )^{2} a^{2} x^{2}-9 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a x -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 ) a \,c^{3}}-\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} \left (a^{2} x^{2}-1\right ) a}+\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} \left (a^{2} x^{2}-1\right ) a}-\frac {\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}+1\right )}{c^{3} \left (a^{2} x^{2}-1\right ) a}-\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} \left (a^{2} x^{2}-1\right ) a}+\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} \left (a^{2} x^{2}-1\right ) a}+\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} \left (a^{2} x^{2}-1\right ) a}-\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} \left (a^{2} x^{2}-1\right ) a}+\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 (a x +\sqrt {a x -1}\, \sqrt {a x +1}+1\right )}{c^{3} \left (a^{2} x^{2}-1\right ) a}+\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} \left (a^{2} x^{2}-1\right ) a}-\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} \left (a^{2} x^{2}-1\right ) a}\)	\(955\)

Input:

int(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-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*arccosh(a*x)^2*a^2*x^2-9*arccosh(a*x)*(a*x-1 
)^(1/2)*(a*x+1)^(1/2)*a*x-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)/a/c^3-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2 
)/c^3/(a^2*x^2-1)/a*ln((a*x-1)^(1/2)*(a*x+1)^(1/2)+a*x-1)+2*(a*x+1)^(1/2)* 
(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/(a^2*x^2-1)/a*ln(a*x+(a*x-1)^(1/2 
)*(a*x+1)^(1/2))-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/(a 
^2*x^2-1)/a*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)+1)-4/3*(a*x+1)^(1/2)*(a*x-1 
)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/(a^2*x^2-1)/a*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^2*x^2-1)/a*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^2*x^2-1)/a*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^2*x^2-1)/a*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^2*x^2-1)/a*arccosh(a*x)^2*ln( 
a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)+1)+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2* 
x^2-1))^(1/2)/c^3/(a^2*x^2-1)/a*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/...
 

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(acosh(a*x)**3/(-a**2*c*x**2+c)**(5/2),x)
 

Output:

Integral(acosh(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x)^3/(-a^2*c*x^2 + c)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \] Input:

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2),x)
 

Output:

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {\mathit {acosh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{2}} \] Input:

int(acosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

int(acosh(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**4*x**4 - 2*sqrt( - a**2*x**2 
+ 1)*a**2*x**2 + sqrt( - a**2*x**2 + 1)),x)/(sqrt(c)*c**2)