\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^2 (d-c^2 d x^2)^{3/2}} \, dx\) [199]

Optimal result

Integrand size = 29, antiderivative size = 341 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \] Output:

-(a+b*arccosh(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccosh(c*x))^ 
2/d/(-c^2*d*x^2+d)^(1/2)+2*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x) 
)^2/d/(-c^2*d*x^2+d)^(1/2)-4*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh( 
c*x))*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)- 
4*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/ 
2)*(c*x+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2)-b^2*c*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d/(-c^2*d*x^2+d)^(1/2) 
-b^2*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1 
/2))^2)/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.37 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a b \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)+b^2 \text {arccosh}(c x)^2+a \left (-a+2 a c^2 x^2-2 b c x \sqrt {-1+c x} \sqrt {1+c x} \log (c x)-b c x \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )\right )}{d x \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(2*a*b*(-1 + 2*c^2*x^2)*ArcCosh[c*x] + b^2*ArcCosh[c*x]^2 + a*(-a + 2*a*c^ 
2*x^2 - 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[c*x] - b*c*x*Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2]))/(d*x*Sqrt[d - c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.44 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.82, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {6347, 25, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 6331, 5984, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6314

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6328

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6331

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {4 i b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+2 c^2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a + b*ArcCosh[c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + ((4*I)*b*c*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]*(I*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])] + 
(I/4)*b*PolyLog[2, -E^(2*ArcCosh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcCosh[ 
c*x])]))/(d*Sqrt[d - c^2*d*x^2]) + 2*c^2*((x*(a + b*ArcCosh[c*x])^2)/(d*Sq 
rt[d - c^2*d*x^2]) + ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(((-1/2*I)*(a + 
 b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2*((a + b*ArcCosh[c*x])*Log[1 - E^(2*Arc 
Cosh[c*x])]) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/4)))/(c*d*Sqrt[d - c^2*d 
*x^2]))
 

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 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 4670

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

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

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 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 6331

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[-d^(-1)   Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x 
, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IG 
tQ[n, 0]
 

rule 6347

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 + 1)*((a + 
b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1270\) vs. \(2(359)=718\).

Time = 0.67 (sec) , antiderivative size = 1271, normalized size of antiderivative = 3.73

method	result	size
default	\(\text {Expression too large to display}\)	\(1271\)
parts	\(\text {Expression too large to display}\)	\(1271\)

Input:

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

Output:

a^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2))-b^2*(-4*a 
rccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-4*arccosh(c*x)*l 
n(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-4*(c*x-1)^(1/2)*(c*x+1)^(1/2) 
*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+4*arccosh( 
c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^4*c^4+4*arccosh(c*x)*ln(1 
+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^4*c^4+4*arccosh(c*x)*ln(1-c*x-(c*x-1)^ 
(1/2)*(c*x+1)^(1/2))*x^4*c^4-4*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1 
)^(1/2))^2)*x^2*c^2-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,-(c*x+(c*x-1)^ 
(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+arccosh(c*x)^2-4*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^3*c^3-4*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^3*c^3+(c*x-1)^(1/2 
)*(c*x+1)^(1/2)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c+2*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x*c+2* 
(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x*c 
+2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^4*c^4+4*polylog(2,-c* 
x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^4*c^4+4*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))*x^4*c^4-2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^2*c^ 
2-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1 
)^(1/2))*x^3*c^3-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*ln(1-c*x-(c*x- 
1)^(1/2)*(c*x+1)^(1/2))*x^3*c^3+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x 
**4 - sqrt( - c**2*x**2 + 1)*x**2),x)*a*b*x - sqrt( - c**2*x**2 + 1)*int(a 
cosh(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*x**4 - sqrt( - c**2*x**2 + 1)*x* 
*2),x)*b**2*x + 2*a**2*c**2*x**2 - a**2)/(sqrt(d)*sqrt( - c**2*x**2 + 1)*d 
*x)