\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) [278]

Optimal result

Integrand size = 28, antiderivative size = 541 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=2 b^2 c^2 d \sqrt {d+c^2 d x^2}-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^3 d x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] 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^2*x^2 
+1)^(1/2)-3*b^2*c^3*d*x*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2) 
-b*c*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/x/(c^2*x^2+1)^(1/2)+b*c^3*d* 
x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+3/2*c^2*d*(c^2* 
d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2-1/2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c 
*x))^2/x^2-3*c^2*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2*arctanh(c*x+(c 
^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-b^2*c^2*d*(c^2*d*x^2+d)^(1/2)*arctanh(( 
c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-3*b*c^2*d*(c^2*d*x^2+d)^(1/2)*(a+b*arc 
sinh(c*x))*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)+3*b*c^2*d*( 
c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,c*x+(c^2*x^2+1)^(1/2))/(c^ 
2*x^2+1)^(1/2)+3*b^2*c^2*d*(c^2*d*x^2+d)^(1/2)*polylog(3,-c*x-(c^2*x^2+1)^ 
(1/2))/(c^2*x^2+1)^(1/2)-3*b^2*c^2*d*(c^2*d*x^2+d)^(1/2)*polylog(3,c*x+(c^ 
2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 7.68 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.43 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(a^2*c^2*d - (a^2*d)/(2*x^2))*Sqrt[d*(1 + c^2*x^2)] + (3*a^2*c^2*d^(3/2)*L 
og[x])/2 - (3*a^2*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/2 + 
(2*a*b*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x 
] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-Arc 
Sinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x] 
)]))/Sqrt[1 + c^2*x^2] + b^2*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(2 - (2*c*x*ArcSi 
nh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x]^2 + (ArcSinh[c*x]^2*(Log[1 - E^( 
-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*ArcS 
inh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])])) 
/Sqrt[1 + c^2*x^2] + (2*(PolyLog[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-A 
rcSinh[c*x])]))/Sqrt[1 + c^2*x^2]) + (a*b*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(-2* 
Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x 
]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 
 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcS 
inh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(4*Sqrt[1 + c^2 
*x^2]) + (b^2*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c* 
x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E 
^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Ta 
nh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*Ar 
cSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.03 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.64, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6222, 6218, 25, 354, 90, 73, 221, 6221, 2009, 6231, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 6218

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^2 + (b*c*d*Sqrt[d + 
c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/x) + c^2*x*(a + b*ArcSinh[c*x]) + (b*c* 
(-2*Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/2))/Sqrt[1 + c^2*x^ 
2] + (3*c^2*d*(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 - (2*b*c*Sqrt[d 
+ c^2*d*x^2]*(a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/Sqrt[1 + 
c^2*x^2] + (I*Sqrt[d + c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ 
ArcSinh[c*x]] - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x] 
]) + b*PolyLog[3, -E^ArcSinh[c*x]]) + (2*I)*b*(-((a + b*ArcSinh[c*x])*Poly 
Log[2, E^ArcSinh[c*x]]) + b*PolyLog[3, E^ArcSinh[c*x]])))/Sqrt[1 + c^2*x^2 
]))/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

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 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 6218

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

rule 6221

Int[((a_.) + ArcSinh[(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 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6222

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*((a + b*Arc 
Sinh[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*ArcSinh[c*x])^n, x], x] - Simp[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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 6231

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

Time = 1.29 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.72

method	result	size
default	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{c^{2} x^{2}+1}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d \,\operatorname {arcsinh}\left (x c \right ) x}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c}{x \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \,x^{2}}{c^{2} x^{2}+1}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{2 c^{2} x^{2}+2}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{2 \sqrt {c^{2} x^{2}+1}}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d}{c^{2} x^{2}+1}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-2 x^{3} c^{3}+3 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-x c \right ) d}{\sqrt {c^{2} x^{2}+1}\, x^{2}}\)	\(933\)
parts	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{c^{2} x^{2}+1}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d \,\operatorname {arcsinh}\left (x c \right ) x}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c}{x \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \,x^{2}}{c^{2} x^{2}+1}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{2 c^{2} x^{2}+2}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{2 \sqrt {c^{2} x^{2}+1}}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d}{c^{2} x^{2}+1}-\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2} d}{\sqrt {c^{2} x^{2}+1}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-2 x^{3} c^{3}+3 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-3 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}-x c \right ) d}{\sqrt {c^{2} x^{2}+1}\, x^{2}}\)	\(933\)

Input:

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

Output:

a^2*(-1/2/d/x^2*(c^2*d*x^2+d)^(5/2)+3/2*c^2*(1/3*(c^2*d*x^2+d)^(3/2)+d*((c 
^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x))))+b^2 
*(d*(c^2*x^2+1))^(1/2)*c^4*d/(c^2*x^2+1)*arcsinh(x*c)^2*x^2-3*b^2*(d*(c^2* 
x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,x*c+(c^2*x^2+1)^(1/2))*c^2*d+3/2 
*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*ln(1-x*c-(c^2* 
x^2+1)^(1/2))*c^2*d-2*b^2*(d*(c^2*x^2+1))^(1/2)*c^3*d/(c^2*x^2+1)^(1/2)*ar 
csinh(x*c)*x-b^2*arcsinh(x*c)*(d*(c^2*x^2+1))^(1/2)*d/x/(c^2*x^2+1)^(1/2)* 
c+2*b^2*(d*(c^2*x^2+1))^(1/2)*c^4*d/(c^2*x^2+1)*x^2+1/2*b^2*(d*(c^2*x^2+1) 
)^(1/2)*c^2*d/(c^2*x^2+1)*arcsinh(x*c)^2-1/2*b^2*arcsinh(x*c)^2*(d*(c^2*x^ 
2+1))^(1/2)*d/x^2/(c^2*x^2+1)+3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2 
)*arcsinh(x*c)*polylog(2,x*c+(c^2*x^2+1)^(1/2))*c^2*d-3/2*b^2*(d*(c^2*x^2+ 
1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*ln(1+x*c+(c^2*x^2+1)^(1/2))*c^2 
*d+3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,-x*c-(c^2*x^2+1 
)^(1/2))*c^2*d-2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arctanh(x*c+( 
c^2*x^2+1)^(1/2))*c^2*d+2*b^2*(d*(c^2*x^2+1))^(1/2)*c^2*d/(c^2*x^2+1)-3*b^ 
2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*polylog(2,-x*c-(c^2 
*x^2+1)^(1/2))*c^2*d+a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x^2*(2*(c 
^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2+3*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^( 
1/2))*x^2*c^2-3*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*x^2*c^2-2*x^3*c^3 
+3*polylog(2,x*c+(c^2*x^2+1)^(1/2))*x^2*c^2-3*polylog(2,-x*c-(c^2*x^2+1...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d*(2*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt(c**2*x**2 + 1)*a** 
2 + 4*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**3,x)*a*b*x**2 + 4*int((sqrt( 
c**2*x**2 + 1)*asinh(c*x))/x,x)*a*b*c**2*x**2 + 2*int((sqrt(c**2*x**2 + 1) 
*asinh(c*x)**2)/x**3,x)*b**2*x**2 + 2*int((sqrt(c**2*x**2 + 1)*asinh(c*x)* 
*2)/x,x)*b**2*c**2*x**2 + 3*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a**2*c**2*x 
**2 - 3*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a**2*c**2*x**2))/(2*x**2)