\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx\) [173]

Optimal result

Integrand size = 29, antiderivative size = 630 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-2*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)+3*a*b*c^3*d*x*(-c^2*d*x^2+d)^(1/2)/(c*x- 
1)^(1/2)/(c*x+1)^(1/2)+3*b^2*c^3*d*x*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/(c* 
x-1)^(1/2)/(c*x+1)^(1/2)-b*c*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/x/( 
c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c^3*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x 
))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/2*c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh 
(c*x))^2-1/2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^2+3*c^2*d*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) 
/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)*arctan((c*x-1) 
^(1/2)*(c*x+1)^(1/2))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3*I*b*c^2*d*(-c^2*d*x^2+ 
d)^(1/2)*(a+b*arccosh(c*x))*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) 
)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3*I*b*c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos 
h(c*x))*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)+3*I*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)*polylog(3,-I*(c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2)))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3*I*b^2*c^2*d*(-c^2*d*x^2+d) 
^(1/2)*polylog(3,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c*x-1)^(1/2)/(c*x+1 
)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5444\) vs. \(2(630)=1260\).

Time = 62.10 (sec) , antiderivative size = 5444, normalized size of antiderivative = 8.64 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.61, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {6343, 25, 6327, 6336, 25, 960, 103, 218, 6341, 2009, 6362, 3042, 4668, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6343

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6336

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 960

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6341

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^2 \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-x) (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\)

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {3}{2} c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-x) (a+b \text {arccosh}(c x))-\frac {a+b \text {arccosh}(c x)}{x}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\sqrt {c x-1} \sqrt {c x+1}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}\)

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^2 + (b*c*d*Sqrt[d - 
c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/x) - c^2*x*(a + b*ArcCosh[c*x]) + b*c*( 
Sqrt[-1 + c*x]*Sqrt[1 + c*x] + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])))/(Sq 
rt[-1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh 
[c*x])^2 - (2*b*c*Sqrt[d - c^2*d*x^2]*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x])/c + b*x*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2* 
d*x^2]*(2*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]] + (2*I)*b*(-((a + 
b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]]) + b*PolyLog[3, (-I)*E^Arc 
Cosh[c*x]]) - (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]] 
) + b*PolyLog[3, I*E^ArcCosh[c*x]])))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/2
 

Defintions of rubi rules used

rule 25

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

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 2009

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 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 6336

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 6341

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq 
rt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* 
x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x 
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6343

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m 
 + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*( 
m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G 
tQ[p, 0] && LtQ[m, -1]
 

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Subst 
[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[m]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3}}d x\]

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^3,x)
 

Output:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^3,x)
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^3,x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/x^3,x, algorithm="maxi 
ma")
 

Output:

1/2*(3*c^2*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) 
 - (-c^2*d*x^2 + d)^(3/2)*c^2 - 3*sqrt(-c^2*d*x^2 + d)*c^2*d - (-c^2*d*x^2 
 + d)^(5/2)/(d*x^2))*a^2 + integrate((-c^2*d*x^2 + d)^(3/2)*b^2*log(c*x + 
sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^3 + 2*(-c^2*d*x^2 + d)^(3/2)*a*b*log(c*x 
+ sqrt(c*x + 1)*sqrt(c*x - 1))/x^3, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\frac {\sqrt {d}\, d \left (-8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a^{2}+16 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}-16 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}+9 a^{2} c^{2} x^{2}\right )}{8 x^{2}} \] Input:

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

Output:

(sqrt(d)*d*( - 8*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 4*sqrt( - c**2*x* 
*2 + 1)*a**2 + 16*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**3,x)*a*b*x**2 
 - 16*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x,x)*a*b*c**2*x**2 + 8*int(( 
sqrt( - c**2*x**2 + 1)*acosh(c*x)**2)/x**3,x)*b**2*x**2 - 8*int((sqrt( - c 
**2*x**2 + 1)*acosh(c*x)**2)/x,x)*b**2*c**2*x**2 - 12*log(tan(asin(c*x)/2) 
)*a**2*c**2*x**2 + 9*a**2*c**2*x**2))/(8*x**2)