\(\int (c-a^2 c x^2)^{3/2} \text {arccosh}(a x)^3 \, dx\) [65]

Optimal result

Integrand size = 22, antiderivative size = 387 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=-\frac {45 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt {c-a^2 c x^2}}{128 a}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-45/128*a*c*x^2*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/128*c*( 
a*x-1)^(3/2)*(a*x+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a+45/64*c*x*(-a^2*c*x^2+c) 
^(1/2)*arccosh(a*x)+3/32*c*x*(-a*x+1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2)*arccosh 
(a*x)+27/128*c*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^2/a/(a*x-1)^(1/2)/(a*x+1) 
^(1/2)-9/16*a*c*x^2*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^2/(a*x-1)^(1/2)/(a*x 
+1)^(1/2)+3/16*c*(a*x-1)^(3/2)*(a*x+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)*arccosh( 
a*x)^2/a+3/8*c*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^3+1/4*x*(-a^2*c*x^2+c)^ 
(3/2)*arccosh(a*x)^3-3/32*c*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^4/a/(a*x-1)^ 
(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.38 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (96 \text {arccosh}(a x)^4-3 (-64 \cosh (2 \text {arccosh}(a x))+\cosh (4 \text {arccosh}(a x)))-24 \text {arccosh}(a x)^2 (-16 \cosh (2 \text {arccosh}(a x))+\cosh (4 \text {arccosh}(a x)))+12 \text {arccosh}(a x) (-32 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x)))+32 \text {arccosh}(a x)^3 (-8 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x)))\right )}{1024 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

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

Output:

-1/1024*(c*Sqrt[c - a^2*c*x^2]*(96*ArcCosh[a*x]^4 - 3*(-64*Cosh[2*ArcCosh[ 
a*x]] + Cosh[4*ArcCosh[a*x]]) - 24*ArcCosh[a*x]^2*(-16*Cosh[2*ArcCosh[a*x] 
] + Cosh[4*ArcCosh[a*x]]) + 12*ArcCosh[a*x]*(-32*Sinh[2*ArcCosh[a*x]] + Si 
nh[4*ArcCosh[a*x]]) + 32*ArcCosh[a*x]^3*(-8*Sinh[2*ArcCosh[a*x]] + Sinh[4* 
ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
 

Rubi [A] (verified)

Time = 4.65 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {6312, 25, 6310, 6298, 6308, 6327, 6329, 6313, 25, 82, 244, 2009, 6311, 15, 6308, 6354, 15, 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.

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

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 6313

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6311

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

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

Input:

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

Output:

(x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3)/4 + (3*c*((x*Sqrt[c - a^2*c*x^2]* 
ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(8*a*Sqrt[-1 + a* 
x]*Sqrt[1 + a*x]) - (3*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcCosh[a*x]^2)/2 - a*( 
-1/4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a^2) + ArcCo 
sh[a*x]^2/(4*a^3))))/(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])))/4 - (3*a*c*Sqrt[c 
- a^2*c*x^2]*(-1/4*((1 - a^2*x^2)^2*ArcCosh[a*x]^2)/a^2 + ((a*(x^2/2 - (a^ 
2*x^4)/4))/4 + (x*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/4 - (3*(- 
1/4*(a*x^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/2 - ArcCosh[a* 
x]^2/(4*a)))/4)/(2*a)))/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 25

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

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 6298

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

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 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(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]
 

rule 6311

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqr 
t[(d2_) + (e2_.)*(x_)], x_Symbol] :> Simp[x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x] 
*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[(1/2)*Simp[Sqrt[d1 + e1*x]/Sqrt[1 
+ c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]   Int[(a + b*ArcCosh[c*x])^n/( 
Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d1 + e1*x] 
/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]   Int[x*(a + b*ArcCosh 
[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c 
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0]
 

rule 6312

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*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] && GtQ[p, 0]
 

rule 6313

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[x*(d1 + e1*x)^p*(d2 + e2*x)^p 
*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d1*d2*(p/(2*p + 1))   Int 
[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - S 
imp[b*c*(n/(2*p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2}, x] && EqQ[e1, c 
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && GtQ[p, 0]
 

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 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 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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{4} c}{32 \sqrt {a x -1}\, \sqrt {a x +1}\, a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )-3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{3} x^{3}-\sqrt {a x -1}\, \sqrt {a x +1}+4 a x \right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )+3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}\)	\(536\)

Input:

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

Output:

-3/32*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4* 
c-1/2048*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-12*a^3*x^3+8*(a*x-1)^(1/2)*(a*x 
+1)^(1/2)*a^4*x^4+4*a*x-8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^(1/2 
)*(a*x+1)^(1/2))*(32*arccosh(a*x)^3-24*arccosh(a*x)^2+12*arccosh(a*x)-3)*c 
/(a*x-1)/(a*x+1)/a+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2* 
(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3 
-6*arccosh(a*x)^2+6*arccosh(a*x)-3)*c/(a*x-1)/(a*x+1)/a+1/32*(-c*(a^2*x^2- 
1))^(1/2)*(-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a^3*x^3+(a*x-1)^(1/2)* 
(a*x+1)^(1/2)-2*a*x)*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)* 
c/(a*x-1)/(a*x+1)/a-1/2048*(-c*(a^2*x^2-1))^(1/2)*(-8*(a*x-1)^(1/2)*(a*x+1 
)^(1/2)*a^4*x^4+8*a^5*x^5+8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-12*a^3*x^3 
-(a*x-1)^(1/2)*(a*x+1)^(1/2)+4*a*x)*(32*arccosh(a*x)^3+24*arccosh(a*x)^2+1 
2*arccosh(a*x)+3)*c/(a*x-1)/(a*x+1)/a
 

Fricas [F]

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

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

Output:

integral(-(a^2*c*x^2 - c)*sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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