3.1.79 \(\int x^3 \text {arccosh}(a x)^{3/2} \, dx\) [79]

3.1.79.1 Optimal result

Integrand size = 12, antiderivative size = 209 \[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}}{32 a}-\frac {3 \text {arccosh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{128 a^4} \]

output

-3/32*arccosh(a*x)^(3/2)/a^4+1/4*x^4*arccosh(a*x)^(3/2)-3/256*erf(2^(1/2)* 
arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4+3/256*erfi(2^(1/2)*arccosh(a*x)^( 
1/2))*2^(1/2)*Pi^(1/2)/a^4-3/2048*erf(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4+3 
/2048*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)/a^4-9/64*x*(a*x-1)^(1/2)*(a*x+1) 
^(1/2)*arccosh(a*x)^(1/2)/a^3-3/32*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh 
(a*x)^(1/2)/a
 

3.1.79.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.48 \[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=\frac {\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-4 \text {arccosh}(a x)\right )+8 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {5}{2},-2 \text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (8 \sqrt {2} \Gamma \left (\frac {5}{2},2 \text {arccosh}(a x)\right )+\Gamma \left (\frac {5}{2},4 \text {arccosh}(a x)\right )\right )}{512 a^4 \sqrt {\text {arccosh}(a x)}} \]

input

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

output

(Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -4*ArcCosh[a*x]] + 8*Sqrt[2]*Sqrt[-ArcCosh 
[a*x]]*Gamma[5/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(8*Sqrt[2]*Gamma[5 
/2, 2*ArcCosh[a*x]] + Gamma[5/2, 4*ArcCosh[a*x]]))/(512*a^4*Sqrt[ArcCosh[a 
*x]])
 

3.1.79.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6299, 6354, 6302, 5971, 2009, 6354, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}

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.

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \int \frac {x^2 \sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {\text {arccosh}(a x)}}dx}{4 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (-\frac {\int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (-\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (-\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (-\frac {\int -\frac {i \sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {i \int \frac {\sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {i \left (i \int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^{3/2}-\frac {3}{8} a \left (-\frac {-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{4 a^2}+\frac {3 \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a^3}+\frac {\text {arccosh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}{2 a^2}\right )}{4 a^2}\right )\)

input

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

output

(x^4*ArcCosh[a*x]^(3/2))/4 - (3*a*((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ 
ArcCosh[a*x]])/(4*a^2) - (-1/32*(Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]]) - (Sq 
rt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCos 
h[a*x]]])/32 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8)/(8*a^5) + 
(3*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(2*a^2) + ArcCosh[ 
a*x]^(3/2)/(3*a^3) + ((I/8)*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[ 
a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/a^3))/(4*a^2) 
))/8
 

3.1.79.3.1 Defintions of rubi rules used

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3789

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6299

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
 

rule 6302

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

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

3.1.79.4 Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-32 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+24 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \sqrt {2}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-3 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{256 \sqrt {\pi }\, a^{4}}-\frac {-512 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}+192 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+512 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-96 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -64 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-3 \pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{2048 \sqrt {\pi }\, a^{4}}\)	\(242\)

input

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

output

-1/256*2^(1/2)*(-32*2^(1/2)*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2+24*2^(1/2) 
*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+16*2^(1/2)*ar 
ccosh(a*x)^(3/2)*Pi^(1/2)+3*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-3*Pi*erfi(2 
^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4-1/2048*(-512*arccosh(a*x)^(3/2)*P 
i^(1/2)*a^4*x^4+192*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2 
)*a^3*x^3+512*arccosh(a*x)^(3/2)*Pi^(1/2)*a^2*x^2-96*arccosh(a*x)^(1/2)*Pi 
^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-64*arccosh(a*x)^(3/2)*Pi^(1/2)+3*Pi 
*erf(2*arccosh(a*x)^(1/2))-3*Pi*erfi(2*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4
 

3.1.79.5 Fricas [F(-2)]

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

input

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

output

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

3.1.79.6 Sympy [F]

\[ \int x^3 \text {arccosh}(a x)^{3/2} \, dx=\int x^{3} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

input

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

output

Integral(x**3*acosh(a*x)**(3/2), x)
 

3.1.79.7 Maxima [F]

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

input

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

output

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

3.1.79.8 Giac [F(-2)]

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

input

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

3.1.79.9 Mupad [F(-1)]

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

input

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

output

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