3.1.80 \(\int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx\) [80]

3.1.80.1 Optimal result

Integrand size = 12, antiderivative size = 514 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=-\frac {3 b^2 \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right )}-\frac {3 b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^3}{2 x^2}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{1+a^2}+\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {3 b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{1+a^2}-\frac {3 a b^2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {3 a b^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \]

output

-3/2*b^2*arcsinh(b*x+a)^2/(a^2+1)-1/2*arcsinh(b*x+a)^3/x^2+3*b^2*arcsinh(b 
*x+a)*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)+3/2*a*b^ 
2*arcsinh(b*x+a)^2*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^ 
2+1)^(3/2)+3*b^2*arcsinh(b*x+a)*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1 
)^(1/2)))/(a^2+1)-3/2*a*b^2*arcsinh(b*x+a)^2*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/ 
2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(3/2)+3*b^2*polylog(2,(b*x+a+(1+(b*x+a)^2)^ 
(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)+3*a*b^2*arcsinh(b*x+a)*polylog(2,(b*x+a+ 
(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^(3/2)+3*b^2*polylog(2,(b*x 
+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)-3*a*b^2*arcsinh(b*x+a)* 
polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(3/2)-3*a 
*b^2*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^(3/2 
)+3*a*b^2*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1) 
^(3/2)-3/2*b*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/(a^2+1)/x
 

3.1.80.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.02 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\frac {-3 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x)^2-3 \sqrt {1+a^2} b x \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2-\sqrt {1+a^2} \text {arcsinh}(a+b x)^3-a^2 \sqrt {1+a^2} \text {arcsinh}(a+b x)^3+6 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-3 a b^2 x^2 \text {arcsinh}(a+b x)^2 \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+6 \sqrt {1+a^2} b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+3 a b^2 x^2 \text {arcsinh}(a+b x)^2 \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+6 b^2 x^2 \left (\sqrt {1+a^2}+a \text {arcsinh}(a+b x)\right ) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 b^2 x^2 \left (\sqrt {1+a^2}-a \text {arcsinh}(a+b x)\right ) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-6 a b^2 x^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 a b^2 x^2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2} x^2} \]

input

Integrate[ArcSinh[a + b*x]^3/x^3,x]
 

output

(-3*Sqrt[1 + a^2]*b^2*x^2*ArcSinh[a + b*x]^2 - 3*Sqrt[1 + a^2]*b*x*Sqrt[1 
+ a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2 - Sqrt[1 + a^2]*ArcSinh[a + 
b*x]^3 - a^2*Sqrt[1 + a^2]*ArcSinh[a + b*x]^3 + 6*Sqrt[1 + a^2]*b^2*x^2*Ar 
cSinh[a + b*x]*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + 
a^2])] - 3*a*b^2*x^2*ArcSinh[a + b*x]^2*Log[(a + Sqrt[1 + a^2] - E^ArcSinh 
[a + b*x])/(a + Sqrt[1 + a^2])] + 6*Sqrt[1 + a^2]*b^2*x^2*ArcSinh[a + b*x] 
*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] + 3*a 
*b^2*x^2*ArcSinh[a + b*x]^2*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/ 
(-a + Sqrt[1 + a^2])] + 6*b^2*x^2*(Sqrt[1 + a^2] + a*ArcSinh[a + b*x])*Pol 
yLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*b^2*x^2*(Sqrt[1 + a^2] 
 - a*ArcSinh[a + b*x])*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] 
- 6*a*b^2*x^2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*a*b^2 
*x^2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(2*(1 + a^2)^(3/2 
)*x^2)
 

3.1.80.3 Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.87, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.583, Rules used = {6274, 25, 27, 6243, 6258, 3042, 3805, 3042, 3803, 2694, 27, 2620, 3011, 2720, 6095, 2620, 2715, 2838, 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.

\(\displaystyle \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)^3}{x^3}d(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\text {arcsinh}(a+b x)^3}{x^3}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -b^2 \int -\frac {\text {arcsinh}(a+b x)^3}{b^3 x^3}d(a+b x)\)

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

\(\displaystyle -b^2 \left (\frac {\text {arcsinh}(a+b x)^3}{2 b^2 x^2}-\frac {3}{2} \int \frac {\text {arcsinh}(a+b x)^2}{(a+i \sin (i \text {arcsinh}(a+b x)))^2}d\text {arcsinh}(a+b x)\right )\)

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSinh[a + b*x]^3/x^3,x]
 

output

-(b^2*(ArcSinh[a + b*x]^3/(2*b^2*x^2) - (3*(-((Sqrt[1 + (a + b*x)^2]*ArcSi 
nh[a + b*x]^2)/((1 + a^2)*b*x)) - (2*(ArcSinh[a + b*x]^2/2 - ArcSinh[a + b 
*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - ArcSinh[a + b*x]*Log 
[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - PolyLog[2, E^ArcSinh[a + b* 
x]/(a - Sqrt[1 + a^2])] - PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2] 
)]))/(1 + a^2) + (2*a*(-1/2*(-(ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b* 
x]/(a - Sqrt[1 + a^2])]) + 2*(-(ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + 
b*x]/(a - Sqrt[1 + a^2])]) + PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a 
^2])]))/Sqrt[1 + a^2] + (-(ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/( 
a + Sqrt[1 + a^2])]) + 2*(-(ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x] 
/(a + Sqrt[1 + a^2])]) + PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2]) 
]))/(2*Sqrt[1 + a^2])))/(1 + a^2)))/2))
 

3.1.80.3.1 Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 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 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6243

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

rule 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b 
, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

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]
 

3.1.80.4 Maple [F]

\[\int \frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{x^{3}}d x\]

input

int(arcsinh(b*x+a)^3/x^3,x)
 

output

int(arcsinh(b*x+a)^3/x^3,x)
 

3.1.80.5 Fricas [F]

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

input

integrate(arcsinh(b*x+a)^3/x^3,x, algorithm="fricas")
 

output

integral(arcsinh(b*x + a)^3/x^3, x)
 

3.1.80.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^3} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{x^{3}}\, dx \]

input

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

output

Integral(asinh(a + b*x)**3/x**3, x)
 

3.1.80.7 Maxima [F]

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

input

integrate(arcsinh(b*x+a)^3/x^3,x, algorithm="maxima")
 

output

-1/2*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3/x^2 + integrate(3/ 
2*(b^3*x^2 + 2*a*b^2*x + a^2*b + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x 
+ a*b) + b)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/(b^3*x^5 + 
3*a*b^2*x^4 + (3*a^2*b + b)*x^3 + (a^3 + a)*x^2 + (b^2*x^4 + 2*a*b*x^3 + ( 
a^2 + 1)*x^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)), x)
 

3.1.80.8 Giac [F]

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

input

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

output

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

3.1.80.9 Mupad [F(-1)]

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

input

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

output

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