3.3.24 \(\int \frac {\text {arccosh}(a x)^2}{(c-a^2 c x^2)^{7/2}} \, dx\) [224]

3.3.24.1 Optimal result

Integrand size = 22, antiderivative size = 429 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}-\frac {x}{30 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]

output

1/5*x*arccosh(a*x)^2/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arccosh(a*x)^2/c^2/(-a^ 
2*c*x^2+c)^(3/2)-1/3*x/c^3/(-a^2*c*x^2+c)^(1/2)-1/30*x/c^3/(-a*x+1)/(a*x+1 
)/(-a^2*c*x^2+c)^(1/2)+8/15*x*arccosh(a*x)^2/c^3/(-a^2*c*x^2+c)^(1/2)+1/10 
*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1)^2/(-a^2*c*x^2 
+c)^(1/2)+4/15*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^3/(-a^2*x^2+1) 
/(-a^2*c*x^2+c)^(1/2)+8/15*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^ 
3/(-a^2*c*x^2+c)^(1/2)-16/15*arccosh(a*x)*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^3/(-a^2*c*x^2+c)^(1/2)-8/15*poly 
log(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 
^3/(-a^2*c*x^2+c)^(1/2)
 

3.3.24.2 Mathematica [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.51 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {a x \left (10+\frac {1}{1-a^2 x^2}\right )+2 \left (8 \sqrt {\frac {-1+a x}{1+a x}}+a x \left (-8+8 \sqrt {\frac {-1+a x}{1+a x}}-\frac {3}{\left (-1+a^2 x^2\right )^2}+\frac {4}{-1+a^2 x^2}\right )\right ) \text {arccosh}(a x)^2+\frac {\left (\frac {-1+a x}{1+a x}\right )^{3/2} \text {arccosh}(a x) \left (-11+8 a^2 x^2+32 \left (-1+a^2 x^2\right )^2 \log \left (1-e^{-2 \text {arccosh}(a x)}\right )\right )}{(-1+a x)^3}-16 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(a x)}\right )}{30 a c^3 \sqrt {c-a^2 c x^2}} \]

input

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

output

-1/30*(a*x*(10 + (1 - a^2*x^2)^(-1)) + 2*(8*Sqrt[(-1 + a*x)/(1 + a*x)] + a 
*x*(-8 + 8*Sqrt[(-1 + a*x)/(1 + a*x)] - 3/(-1 + a^2*x^2)^2 + 4/(-1 + a^2*x 
^2)))*ArcCosh[a*x]^2 + (((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]*(-11 + 8 
*a^2*x^2 + 32*(-1 + a^2*x^2)^2*Log[1 - E^(-2*ArcCosh[a*x])]))/(-1 + a*x)^3 
 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*PolyLog[2, E^(-2*ArcCosh[a*x])] 
)/(a*c^3*Sqrt[c - a^2*c*x^2])
 

3.3.24.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.37 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.95, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {6316, 25, 6316, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 6329, 41, 42, 41}

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.

\(\displaystyle \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6316

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

\(\Big \downarrow \) 6314

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6328

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)}{\left (1-a^2 x^2\right )^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \log \left (1-e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)}{\left (1-a^2 x^2\right )^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \int \frac {x \text {arccosh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6329

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\int \frac {1}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 a}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\int \frac {1}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{2 a}+\frac {\text {arccosh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 41

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\int \frac {1}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 a}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {x}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {-\frac {2}{3} \int \frac {1}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {x}{3 (a x-1)^{3/2} (a x+1)^{3/2}}}{4 a}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {x}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 41

\(\displaystyle \frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\frac {2 x}{3 \sqrt {a x-1} \sqrt {a x+1}}-\frac {x}{3 (a x-1)^{3/2} (a x+1)^{3/2}}}{4 a}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \left (\frac {\text {arccosh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {x}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)^2}{c \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x) \log \left (1-e^{2 \text {arccosh}(a x)}\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \text {arccosh}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arccosh}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

input

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

output

(x*ArcCosh[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (2*a*Sqrt[-1 + a*x]*Sqrt[ 
1 + a*x]*(-1/4*(-1/3*x/((-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) + (2*x)/(3*Sqrt[ 
-1 + a*x]*Sqrt[1 + a*x]))/a + ArcCosh[a*x]/(4*a^2*(1 - a^2*x^2)^2)))/(5*c^ 
3*Sqrt[c - a^2*c*x^2]) + (4*((x*ArcCosh[a*x]^2)/(3*c*(c - a^2*c*x^2)^(3/2) 
) + (2*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-1/2*x/(a*Sqrt[-1 + a*x]*Sqrt[1 + a 
*x]) + ArcCosh[a*x]/(2*a^2*(1 - a^2*x^2))))/(3*c^2*Sqrt[c - a^2*c*x^2]) + 
(2*((x*ArcCosh[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) + ((2*I)*Sqrt[-1 + a*x]*Sqr 
t[1 + a*x]*((-1/2*I)*ArcCosh[a*x]^2 - (2*I)*(-1/2*(ArcCosh[a*x]*Log[1 - E^ 
(2*ArcCosh[a*x])]) - PolyLog[2, E^(2*ArcCosh[a*x])]/4)))/(a*c*Sqrt[c - a^2 
*c*x^2])))/(3*c)))/(5*c)
 

3.3.24.3.1 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 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- 
x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 
3)/(2*a*c*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
 

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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 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]
 

3.3.24.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(793\) vs. \(2(395)=790\).

Time = 1.23 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.85

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+15 a x +16 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-8 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-64 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{7} x^{7}-64 \,\operatorname {arccosh}\left (a x \right ) a^{8} x^{8}-32 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}-32 a^{8} x^{8}+248 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{5} x^{5}+280 \,\operatorname {arccosh}\left (a x \right ) a^{6} x^{6}+126 x^{5} a^{5} \sqrt {a x -1}\, \sqrt {a x +1}+142 a^{6} x^{6}+80 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}-340 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-456 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-156 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-265 a^{4} x^{4}-190 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}+165 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+328 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+62 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +235 a^{2} x^{2}+128 \operatorname {arccosh}\left (a x \right )^{2}-88 \,\operatorname {arccosh}\left (a x \right )-80\right )}{30 \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) c^{4} a}-\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2}}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}+\frac {16 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}\)	\(794\)

input

int(arccosh(a*x)^2/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/30*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3-8*(a*x-1)^(1/2)*(a*x+1) 
^(1/2)*a^4*x^4+15*a*x+16*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-8*(a*x-1)^(1/ 
2)*(a*x+1)^(1/2))*(-64*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^7*x^7-64 
*arccosh(a*x)*a^8*x^8-32*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^7*x^7-32*a^8*x^8+24 
8*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^5*x^5+280*arccosh(a*x)*a^6*x^ 
6+126*x^5*a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)+142*a^6*x^6+80*a^4*x^4*arccosh(a 
*x)^2-340*a^3*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-456*a^4*x^4*arc 
cosh(a*x)-156*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-265*a^4*x^4-190*a^2*x^2* 
arccosh(a*x)^2+165*a*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+328*a^2*x^ 
2*arccosh(a*x)+62*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+235*a^2*x^2+128*arccosh( 
a*x)^2-88*arccosh(a*x)-80)/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x 
^4+287*a^2*x^2-64)/c^4/a-16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1) 
)^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)^2+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2) 
*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1 
/2)*(a*x+1)^(1/2))+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2 
)/c^4/a/(a^2*x^2-1)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/15*(a*x 
+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a 
*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2 
)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*polylog(2,a*x+(a*x-1)^(1/2)*(a* 
x+1)^(1/2))
 

3.3.24.5 Fricas [F]

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

input

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

output

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^2/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 
+ 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4), x)
 

3.3.24.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]

input

integrate(acosh(a*x)**2/(-a**2*c*x**2+c)**(7/2),x)
 

output

Timed out
 

3.3.24.7 Maxima [F]

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

input

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

output

integrate(arccosh(a*x)^2/(-a^2*c*x^2 + c)^(7/2), x)
 

3.3.24.8 Giac [F(-2)]

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

input

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

3.3.24.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]

input

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

output

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