\(\int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx\) [110]

Optimal result

Integrand size = 12, antiderivative size = 244 \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {4 x^2}{5 a^2 \text {arccosh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arccosh}(a x)^{3/2}}+\frac {16 x \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {128 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{15 a \sqrt {\text {arccosh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{15 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{15 a^4} \] Output:

-2/5*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(5/2)+4/5*x^2/a^2/arcc 
osh(a*x)^(3/2)-16/15*x^4/arccosh(a*x)^(3/2)+16/5*x*(a*x-1)^(1/2)*(a*x+1)^( 
1/2)/a^3/arccosh(a*x)^(1/2)-128/15*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arcco 
sh(a*x)^(1/2)+16/15*Pi^(1/2)*erf(2*arccosh(a*x)^(1/2))/a^4+4/15*2^(1/2)*Pi 
^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a^4+16/15*Pi^(1/2)*erfi(2*arccosh(a 
*x)^(1/2))/a^4+4/15*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arccosh(a*x)^(1/2))/a^4
 

Mathematica [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {e^{-4 \text {arccosh}(a x)} \left (3-3 e^{8 \text {arccosh}(a x)}-8 \text {arccosh}(a x)-8 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)+64 \text {arccosh}(a x)^2-64 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)^2+128 e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-8 e^{2 \text {arccosh}(a x)} \left (3 a e^{2 \text {arccosh}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\text {arccosh}(a x)+e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)-4 \text {arccosh}(a x)^2+4 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^2-4 \sqrt {2} e^{2 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+4 \sqrt {2} e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )-128 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )}{120 a^4 \text {arccosh}(a x)^{5/2}} \] Input:

Integrate[x^3/ArcCosh[a*x]^(7/2),x]
 

Output:

(3 - 3*E^(8*ArcCosh[a*x]) - 8*ArcCosh[a*x] - 8*E^(8*ArcCosh[a*x])*ArcCosh[ 
a*x] + 64*ArcCosh[a*x]^2 - 64*E^(8*ArcCosh[a*x])*ArcCosh[a*x]^2 + 128*E^(4 
*ArcCosh[a*x])*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -4*ArcCosh[a*x]] - 8*E^(2* 
ArcCosh[a*x])*(3*a*E^(2*ArcCosh[a*x])*x*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a* 
x) + ArcCosh[a*x] + E^(4*ArcCosh[a*x])*ArcCosh[a*x] - 4*ArcCosh[a*x]^2 + 4 
*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^2 - 4*Sqrt[2]*E^(2*ArcCosh[a*x])*(-ArcCos 
h[a*x])^(5/2)*Gamma[1/2, -2*ArcCosh[a*x]] + 4*Sqrt[2]*E^(2*ArcCosh[a*x])*A 
rcCosh[a*x]^(5/2)*Gamma[1/2, 2*ArcCosh[a*x]]) - 128*E^(4*ArcCosh[a*x])*Arc 
Cosh[a*x]^(5/2)*Gamma[1/2, 4*ArcCosh[a*x]])/(120*a^4*E^(4*ArcCosh[a*x])*Ar 
cCosh[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6301, 6366, 6300, 25, 2009, 3042, 3788, 26, 2611, 2633, 2634}

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.

\(\displaystyle \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6301

\(\displaystyle \frac {8}{5} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{5/2}}dx-\frac {6 \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{5/2}}dx}{5 a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 6366

\(\displaystyle -\frac {6 \left (\frac {4 \int \frac {x}{\text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \int \frac {x^3}{\text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 6300

\(\displaystyle \frac {8}{5} a \left (\frac {8 \left (-\frac {2 \int \left (-\frac {\cosh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}-\frac {\cosh (4 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {6 \left (\frac {4 \left (-\frac {2 \int -\frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {8}{5} a \left (\frac {8 \left (-\frac {2 \int \left (-\frac {\cosh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}-\frac {\cosh (4 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cosh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {6 \left (-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}+\frac {2 \int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^2}\right )}{3 a}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 3788

\(\displaystyle -\frac {6 \left (-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 \left (\frac {1}{2} i \int \frac {i e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int -\frac {i e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}\right )}{3 a}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2611

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {6 \left (\frac {4 \left (-\frac {2 \left (-\int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {8}{5} a \left (\frac {8 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {6 \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

Input:

Int[x^3/ArcCosh[a*x]^(7/2),x]
 

Output:

(-2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) - (6*((-2*x 
^2)/(3*a*ArcCosh[a*x]^(3/2)) + (4*((-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a* 
Sqrt[ArcCosh[a*x]]) - (2*(-1/2*(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] 
) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/2))/a^2))/(3*a)))/(5*a) 
+ (8*a*((-2*x^4)/(3*a*ArcCosh[a*x]^(3/2)) + (8*((-2*x^3*Sqrt[-1 + a*x]*Sqr 
t[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) - (2*(-1/8*(Sqrt[Pi]*Erf[2*Sqrt[ArcCosh 
[a*x]]]) - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4 - (Sqrt[Pi]*Erfi 
[2*Sqrt[ArcCosh[a*x]]])/8 - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/ 
4))/a^4))/(3*a)))/5
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 6300

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + Simp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

rule 6301

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + (-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCosh[c*x 
])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) 
 Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) 
), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
 

rule 6366

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 
_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a 
 + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x 
]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp 
[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Int[ 
(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.50

method	result	size
default	\(-\frac {\sqrt {2}\, \left (16 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +4 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }-8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-8 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{30 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{3}}-\frac {128 \sqrt {\pi }\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} a^{3} x^{3}+16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}-64 \sqrt {\pi }\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} a x +6 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-16 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-3 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -16 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-16 \operatorname {arccosh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+2 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }}{15 \sqrt {\pi }\, a^{4} \operatorname {arccosh}\left (a x \right )^{3}}\)	\(366\)

Input:

int(x^3/arccosh(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30*2^(1/2)*(16*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1 
)^(1/2)*a*x+4*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+3*2^(1/2)*arccos 
h(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-2*arccosh(a*x)^(3/2) 
*2^(1/2)*Pi^(1/2)-8*arccosh(a*x)^3*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-8*ar 
ccosh(a*x)^3*Pi*erfi(2^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4/arccosh(a*x 
)^3-1/15*(128*Pi^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(5/2)*a^3* 
x^3+16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^4*x^4-64*Pi^(1/2)*(a*x-1)^(1/2)*(a*x+ 
1)^(1/2)*arccosh(a*x)^(5/2)*a*x+6*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2 
)*(a*x-1)^(1/2)*a^3*x^3-16*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2-3*arccosh(a 
*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-16*arccosh(a*x)^3*Pi*er 
f(2*arccosh(a*x)^(1/2))-16*arccosh(a*x)^3*Pi*erfi(2*arccosh(a*x)^(1/2))+2* 
arccosh(a*x)^(3/2)*Pi^(1/2))/Pi^(1/2)/a^4/arccosh(a*x)^3
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/arccosh(a*x)^(7/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(x**3/acosh(a*x)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^3/arccosh(a*x)^(7/2),x, algorithm="maxima")
 

Output:

integrate(x^3/arccosh(a*x)^(7/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/arccosh(a*x)^(7/2),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. \[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^3/acosh(a*x)^(7/2),x)
 

Output:

int(x^3/acosh(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^3}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {\frac {8 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{4}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a^{2}}{5}-\frac {6 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right )}{5}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{5}}{\mathit {acosh} \left (a x \right )^{3} a} \] Input:

int(x^3/acosh(a*x)^(7/2),x)
 

Output:

(2*(4*acosh(a*x)**3*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**4 
)/(acosh(a*x)**3*a**2*x**2 - acosh(a*x)**3),x)*a**2 - 3*acosh(a*x)**3*int( 
(sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**2)/(acosh(a*x)**3*a**2*x* 
*2 - acosh(a*x)**3),x) - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**3 
))/(5*acosh(a*x)**3*a)