3.3.10 \(\int \frac {(a+b \text {arccosh}(c x))^2}{x (d-c^2 d x^2)^{3/2}} \, dx\) [210]

3.3.10.1 Optimal result

Integrand size = 29, antiderivative size = 471 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]

output

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

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

Time = 3.08 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\frac {a^2 \sqrt {d-c^2 d x^2}}{-1+c^2 x^2}-a^2 \sqrt {d} \log (c x)+a^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 i a b d \left (i \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)^2}{1-c x}+2 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+i \text {arccosh}(c x)^2 \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x)^2 \log \left (1+i e^{-\text {arccosh}(c x)}\right )-2 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+2 i \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-2 i \text {arccosh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+2 i \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(c x)}\right )-2 i \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \]

input

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

output

-(((a^2*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2) - a^2*Sqrt[d]*Log[c*x] + a^2*S 
qrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + ((2*I)*a*b*d*(I*ArcCosh[c*x] 
 + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c 
*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCo 
sh[c*x]] + I*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2] 
] - I*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + Sqr 
t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[( 
-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2 
*d*x^2] + (b^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((Sqrt[(-1 + c*x)/(1 
 + c*x)]*ArcCosh[c*x]^2)/(1 - c*x) + 2*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c* 
x])] + I*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]^2*Log[1 
 + I/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 2*PolyL 
og[2, -E^(-ArcCosh[c*x])] + (2*I)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c 
*x]] - (2*I)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] - 2*PolyLog[2, E^(- 
ArcCosh[c*x])] + (2*I)*PolyLog[3, (-I)/E^ArcCosh[c*x]] - (2*I)*PolyLog[3, 
I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2])/d^2)
 

3.3.10.3 Rubi [A] (verified)

Time = 2.33 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.55, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {6351, 25, 6304, 6318, 3042, 26, 4670, 2715, 2838, 6361, 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.

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

\(\Big \downarrow \) 6351

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6304

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

(a + b*ArcCosh[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*Sqrt[-1 + c*x]*S 
qrt[1 + c*x]*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*Pol 
yLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/(d*Sqrt[d - c^ 
2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*(a + b*ArcCosh[c*x])^2*ArcTan 
[E^ArcCosh[c*x]] + (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCo 
sh[c*x]]) + b*PolyLog[3, (-I)*E^ArcCosh[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]])))/( 
d*Sqrt[d - c^2*d*x^2])
 

3.3.10.3.1 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 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 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 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 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 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 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6318

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

rule 6351

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

rule 6361

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

3.3.10.4 Maple [F]

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

input

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

output

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

3.3.10.5 Fricas [F]

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

input

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

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^5 - 2*c^2*d^2*x^3 + d^2*x), x)
 

3.3.10.6 Sympy [F]

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

input

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

output

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

3.3.10.7 Maxima [F]

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

input

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

output

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

3.3.10.8 Giac [F]

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

input

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

3.3.10.9 Mupad [F(-1)]

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

input

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

output

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