\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 (d+c^2 d x^2)^{3/2}} \, dx\) [314]

Optimal result

Integrand size = 28, antiderivative size = 305 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{d x \sqrt {d+c^2 d x^2}}-\frac {2 c^2 x (a+b \text {arcsinh}(c x))^2}{d \sqrt {d+c^2 d x^2}}-\frac {2 c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{d \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {4 b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a^2+2 a^2 c^2 x^2+2 a b \text {arcsinh}(c x)+4 a b c^2 x^2 \text {arcsinh}(c x)+b^2 \text {arcsinh}(c x)^2+2 b^2 c^2 x^2 \text {arcsinh}(c x)^2-2 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-2 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-2 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-2 a b c x \sqrt {1+c^2 x^2} \log (c x)-a b c x \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+b^2 c x \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c x \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{d x \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

-((a^2 + 2*a^2*c^2*x^2 + 2*a*b*ArcSinh[c*x] + 4*a*b*c^2*x^2*ArcSinh[c*x] + 
 b^2*ArcSinh[c*x]^2 + 2*b^2*c^2*x^2*ArcSinh[c*x]^2 - 2*b^2*c*x*Sqrt[1 + c^ 
2*x^2]*ArcSinh[c*x]^2 - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E 
^(-2*ArcSinh[c*x])] - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^( 
-2*ArcSinh[c*x])] - 2*a*b*c*x*Sqrt[1 + c^2*x^2]*Log[c*x] - a*b*c*x*Sqrt[1 
+ c^2*x^2]*Log[1 + c^2*x^2] + b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2* 
ArcSinh[c*x])] + b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])] 
)/(d*x*Sqrt[d + c^2*d*x^2]))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.87, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6224, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6214, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a + b*ArcSinh[c*x])^2/(d*x*Sqrt[d + c^2*d*x^2])) - 2*c^2*((x*(a + b*Arc 
Sinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b*Sqrt[1 + c^2*x^2]*(((-1/2 
*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)*(((a + b*ArcSinh[c*x])*Log[1 + E^(2* 
ArcSinh[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/(c*d*Sqrt[d + 
 c^2*d*x^2])) + ((4*I)*b*c*Sqrt[1 + c^2*x^2]*(I*(a + b*ArcSinh[c*x])*ArcTa 
nh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcSinh[c*x])] - (I/4)*b 
*PolyLog[2, E^(2*ArcSinh[c*x])]))/(d*Sqrt[d + c^2*d*x^2])
 

Defintions of rubi rules used

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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && 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 6202

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, 
 c^2*d] && GtQ[n, 0]
 

rule 6212

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

rule 6214

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

rule 6224

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1086\) vs. \(2(317)=634\).

Time = 1.42 (sec) , antiderivative size = 1087, normalized size of antiderivative = 3.56

method	result	size
default	\(\text {Expression too large to display}\)	\(1087\)
parts	\(\text {Expression too large to display}\)	\(1087\)

Input:

int((a+b*arcsinh(x*c))^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*arcs 
inh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*x^2*c^2+4*arcsinh(x*c)*ln(1+x*c+(c^2* 
x^2+1)^(1/2))*x^2*c^2+4*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c 
^4+4*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*x^4*c^4-4*(c^2*x^2+1)^(1/2)* 
polylog(2,x*c+(c^2*x^2+1)^(1/2))*x^3*c^3-2*(c^2*x^2+1)^(1/2)*polylog(2,-(x 
*c+(c^2*x^2+1)^(1/2))^2)*x^3*c^3-4*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c 
-(c^2*x^2+1)^(1/2))*x^3*c^3+arcsinh(x*c)^2-2*(c^2*x^2+1)^(1/2)*arcsinh(x*c 
)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x*c-2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln( 
1+x*c+(c^2*x^2+1)^(1/2))*x*c+4*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2 
)*x^2*c^2-4*(c^2*x^2+1)^(1/2)*polylog(2,-x*c-(c^2*x^2+1)^(1/2))*x^3*c^3-2* 
(c^2*x^2+1)^(1/2)*polylog(2,x*c+(c^2*x^2+1)^(1/2))*x*c-(c^2*x^2+1)^(1/2)*p 
olylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)*x*c-2*(c^2*x^2+1)^(1/2)*polylog(2,-x* 
c-(c^2*x^2+1)^(1/2))*x*c+4*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*x^4*c^ 
4-4*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^3*c^3 
-4*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*x^3*c^3-2*(c 
^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*x*c+2*polylog(2,- 
(x*c+(c^2*x^2+1)^(1/2))^2)*x^2*c^2+4*polylog(2,x*c+(c^2*x^2+1)^(1/2))*x^2* 
c^2+4*polylog(2,-x*c-(c^2*x^2+1)^(1/2))*x^2*c^2+4*polylog(2,x*c+(c^2*x^2+1 
)^(1/2))*x^4*c^4+2*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+4*polylog 
(2,-x*c-(c^2*x^2+1)^(1/2))*x^4*c^4)*(2*c^2*x^2+2*(c^2*x^2+1)^(1/2)*x*c+...
 

Fricas [F]

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

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(3/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt(c**2*x**2 + 1)*a**2 + 2*in 
t(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x) 
*a*b*c**2*x**3 + 2*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c* 
*2*x**2 + 1)*x**2),x)*a*b*x + int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**2* 
x**4 + sqrt(c**2*x**2 + 1)*x**2),x)*b**2*c**2*x**3 + int(asinh(c*x)**2/(sq 
rt(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**3*x**3 - 2*a**2*c*x)/(sqrt(d)*d*x*(c**2*x**2 + 1))