Integrand size = 12, antiderivative size = 478 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)^2}{3 x^3}-\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}} \]
-1/3*b^2/(a^2+1)/x-1/3*arcsinh(b*x+a)^2/x^3-a*b^3*ln(x)/(a^2+1)^2-a^2*b^3* 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) ^(5/2)+1/3*b^3*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^2*b^3*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)^(5/2)-1/3*b^3*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^2*b^3*polylog(2,(b*x +a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^(5/2)+1/3*b^3*polylog(2 ,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^(3/2)+a^2*b^3*poly log(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(5/2)-1/3*b^3 *polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(3/2)-1/ 3*b*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/(a^2+1)/x^2+a*b^2*arcsinh(b*x+a)*(1 +(b*x+a)^2)^(1/2)/(a^2+1)^2/x
Result contains complex when optimal does not.
Time = 10.00 (sec) , antiderivative size = 1830, normalized size of antiderivative = 3.83 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx =\text {Too large to display} \]
(-((b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*x^2)) - ArcSinh[a + b*x]^2/x^3 - (b^2*(1 + a^2 - 3*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] ))/((1 + a^2)^2*x) + (I*b^3*Pi*ArcTanh[(-1 - a*Tanh[ArcSinh[a + b*x]/2])/S qrt[1 + a^2]])/(1 + a^2)^(5/2) - ((2*I)*a^2*b^3*Pi*ArcTanh[(-1 - a*Tanh[Ar cSinh[a + b*x]/2])/Sqrt[1 + a^2]])/(1 + a^2)^(5/2) - (3*a*b^3*Log[-((b*x)/ a)])/(1 + a^2)^2 + (b^3*(-2*ArcCos[I*a]*ArcTanh[((-I + a)*Cot[(Pi + (2*I)* ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] - (Pi - (2*I)*ArcSinh[a + b*x])*ArcT anh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (ArcC os[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sq rt[-1 - a^2]] + (2*I)*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4 ])/Sqrt[-1 - a^2]])*Log[Sqrt[-1 - a^2]/(Sqrt[2]*E^(ArcSinh[a + b*x]/2)*Sqr t[b*x])] + (ArcCos[I*a] - (2*I)*(ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh [a + b*x])/4])/Sqrt[-1 - a^2]] + ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[ a + b*x])/4])/Sqrt[-1 - a^2]]))*Log[(I*Sqrt[-1 - a^2]*E^(ArcSinh[a + b*x]/ 2))/(Sqrt[2]*Sqrt[b*x])] - (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a + I*(-1 + S qrt[-1 - a^2]))*(I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(I + a - Sqrt[ -1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] - (ArcCos[I*a] - (2*I)*Ar cTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log [((I + a)*(a - I*(1 + Sqrt[-1 - a^2]))*(-I + Cot[(Pi + (2*I)*ArcSinh[a ...
Time = 2.63 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.46, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.833, Rules used = {6274, 27, 6243, 6258, 3042, 3806, 26, 3042, 3147, 17, 3805, 3042, 3147, 16, 3803, 2694, 27, 2620, 2715, 2838, 7293, 2009}
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.
| \(\displaystyle \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)^2}{x^4}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^3 \int \frac {\text {arcsinh}(a+b x)^2}{b^4 x^4}d(a+b x)\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \int -\frac {\text {arcsinh}(a+b x)}{b^3 x^3 \sqrt {(a+b x)^2+1}}d(a+b x)-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 6258 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \int -\frac {\text {arcsinh}(a+b x)}{b^3 x^3}d\text {arcsinh}(a+b x)-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \int \frac {\text {arcsinh}(a+b x)}{(a+i \sin (i \text {arcsinh}(a+b x)))^3}d\text {arcsinh}(a+b x)\right )\) |
| \(\Big \downarrow \) 3806 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (-\frac {\int \frac {\sqrt {(a+b x)^2+1}}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \int \frac {\text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {i \int \frac {i (a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (-\frac {\int \frac {\sqrt {(a+b x)^2+1}}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \int \frac {\text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{a^2+1}+\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \int \frac {\text {arcsinh}(a+b x)}{(a+i \sin (i \text {arcsinh}(a+b x)))^2}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\int \frac {\cos (i \text {arcsinh}(a+b x))}{(a+i \sin (i \text {arcsinh}(a+b x)))^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \int \frac {\text {arcsinh}(a+b x)}{(a+i \sin (i \text {arcsinh}(a+b x)))^2}d\text {arcsinh}(a+b x)}{a^2+1}+\frac {\int \frac {1}{(2 a+b x)^2}d(-a-b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \int \frac {\text {arcsinh}(a+b x)}{(a+i \sin (i \text {arcsinh}(a+b x)))^2}d\text {arcsinh}(a+b x)}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )\right )\) |
| \(\Big \downarrow \) 3805 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (-\frac {\int -\frac {\sqrt {(a+b x)^2+1}}{b x}d\text {arcsinh}(a+b x)}{a^2+1}+\frac {a \int -\frac {\text {arcsinh}(a+b x)}{b x}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {a \int \frac {\text {arcsinh}(a+b x)}{a+i \sin (i \text {arcsinh}(a+b x))}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\int \frac {\cos (i \text {arcsinh}(a+b x))}{a+i \sin (i \text {arcsinh}(a+b x))}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )\right )\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {a \int \frac {\text {arcsinh}(a+b x)}{a+i \sin (i \text {arcsinh}(a+b x))}d\text {arcsinh}(a+b x)}{a^2+1}+\frac {\int \frac {1}{2 a+b x}d(-a-b x)}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle b^3 \left (-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {a \int \frac {\text {arcsinh}(a+b x)}{a+i \sin (i \text {arcsinh}(a+b x))}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )\right )\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {2 a \int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{2 e^{\text {arcsinh}(a+b x)} a-e^{2 \text {arcsinh}(a+b x)}+1}d\text {arcsinh}(a+b x)}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {2 a \left (\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{2 \left (a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}\right )}d\text {arcsinh}(a+b x)}{\sqrt {a^2+1}}-\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{2 \left (a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}\right )}d\text {arcsinh}(a+b x)}{\sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {2 a \left (\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}}d\text {arcsinh}(a+b x)}{2 \sqrt {a^2+1}}-\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}}d\text {arcsinh}(a+b x)}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {2 a \left (\frac {\int \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {\int \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {a \left (\frac {2 a \left (\frac {\int e^{-\text {arcsinh}(a+b x)} \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {\int e^{-\text {arcsinh}(a+b x)} \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \frac {(a+b x) \text {arcsinh}(a+b x)}{b^2 x^2}d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}+\frac {a \left (\frac {2 a \left (\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\int \left (\frac {\text {arcsinh}(a+b x)}{b x}+\frac {a \text {arcsinh}(a+b x)}{b^2 x^2}\right )d\text {arcsinh}(a+b x)}{2 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}+\frac {a \left (\frac {2 a \left (\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^3 \left (-\frac {2}{3} \left (\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{2 \left (a^2+1\right ) b^2 x^2}+\frac {a \left (\frac {2 a \left (\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {-\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )}{a^2+1}-\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {\log (2 a+b x)}{a^2+1}\right )}{a^2+1}+\frac {\frac {a^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}-\frac {a \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right ) b x}+\frac {a^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}-\frac {\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\sqrt {a^2+1}}+\frac {\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\sqrt {a^2+1}}+\frac {a \log (-b x)}{a^2+1}}{2 \left (a^2+1\right )}-\frac {1}{2 \left (a^2+1\right ) (2 a+b x)}\right )-\frac {\text {arcsinh}(a+b x)^2}{3 b^3 x^3}\right )\) |
b^3*(-1/3*ArcSinh[a + b*x]^2/(b^3*x^3) - (2*(-1/2*1/((1 + a^2)*(2*a + b*x) ) + (Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(2*(1 + a^2)*b^2*x^2) + (a*(- ((Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*b*x)) + Log[2*a + b*x ]/(1 + a^2) + (2*a*(-1/2*(-(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])])/ 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])])/(2*Sqrt[1 + a^2])))/(1 + a^2)))/(1 + a^2) + (-((a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*b*x)) + (a^2*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x] /(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) - (ArcSinh[a + b*x]*Log[1 - E^ArcSi nh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] - (a^2*ArcSinh[a + b*x]*Lo g[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(3/2) + (ArcSinh[ a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (a*Log[-(b*x)])/(1 + a^2) + (a^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) - PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a ^2])]/Sqrt[1 + a^2] - (a^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2 ])])/(1 + a^2)^(3/2) + PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]/ Sqrt[1 + a^2])/(2*(1 + a^2))))/3)
3.1.74.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*(c + d*x)^m*Cos[e + f*x]*((a + b*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(a^2 - b^2))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m*(a + b*Sin[e + f*x])^(n + 1), x], x] - Simp[b*((n + 2)/((n + 1)*(a^2 - b^2))) Int[(c + d*x)^m*Sin[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x] + Simp [b*d*(m/(f*(n + 1)*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && IGtQ[m, 0]
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]
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])
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]
Time = 0.41 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(b^{3} \left (-\frac {a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{3}+7 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{2} \left (b x +a \right )-3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a \left (b x +a \right )^{2}-3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{4}+9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )-9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}+3 \,\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+2 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}+a^{4}-2 a^{3} \left (b x +a \right )+a^{2} \left (b x +a \right )^{2}-\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a +\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+a^{2}-2 a \left (b x +a \right )+\left (b x +a \right )^{2}}{3 \left (a^{2}+1\right )^{2} b^{3} x^{3}}+\frac {2 a \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{\left (a^{2}+1\right )^{2}}-\frac {a \ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{\left (a^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}\right )\) | \(764\) |
| default | \(b^{3} \left (-\frac {a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{3}+7 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{2} \left (b x +a \right )-3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a \left (b x +a \right )^{2}-3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{4}+9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )-9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}+3 \,\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+2 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}+a^{4}-2 a^{3} \left (b x +a \right )+a^{2} \left (b x +a \right )^{2}-\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a +\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+a^{2}-2 a \left (b x +a \right )+\left (b x +a \right )^{2}}{3 \left (a^{2}+1\right )^{2} b^{3} x^{3}}+\frac {2 a \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{\left (a^{2}+1\right )^{2}}-\frac {a \ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{\left (a^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}\right )\) | \(764\) |
b^3*(-1/3*(a^4*arcsinh(b*x+a)^2-4*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a^3+7 *arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a^2*(b*x+a)-3*arcsinh(b*x+a)*(1+(b*x+a )^2)^(1/2)*a*(b*x+a)^2-3*arcsinh(b*x+a)*a^4+9*arcsinh(b*x+a)*a^3*(b*x+a)-9 *arcsinh(b*x+a)*a^2*(b*x+a)^2+3*arcsinh(b*x+a)*a*(b*x+a)^3+2*a^2*arcsinh(b *x+a)^2+a^4-2*a^3*(b*x+a)+a^2*(b*x+a)^2-arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2) *a+arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*(b*x+a)+arcsinh(b*x+a)^2+a^2-2*a*(b* x+a)+(b*x+a)^2)/(a^2+1)^2/b^3/x^3+2/(a^2+1)^2*a*ln(b*x+a+(1+(b*x+a)^2)^(1/ 2))-1/(a^2+1)^2*a*ln(2*a*(b*x+a+(1+(b*x+a)^2)^(1/2))-(b*x+a+(1+(b*x+a)^2)^ (1/2))^2+1)-1/3/(a^2+1)^(5/2)*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)-b*x-(1+(b*x +a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+1/3/(a^2+1)^(5/2)*arcsinh(b*x+a)*ln(((a^2 +1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))-1/3/(a^2+1)^(5/2)*d ilog(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+1/3/(a^2+1 )^(5/2)*dilog(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))+ 2/3/(a^2+1)^(5/2)*a^2*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^( 1/2))/(a+(a^2+1)^(1/2)))-2/3/(a^2+1)^(5/2)*a^2*arcsinh(b*x+a)*ln(((a^2+1)^ (1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))+2/3/(a^2+1)^(5/2)*a^2*d ilog(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-2/3/(a^2+1 )^(5/2)*a^2*dilog(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2 ))))
\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]
-1/3*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/x^3 + integrate(2/ 3*(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))/(b^3*x^6 + 3* a*b^2*x^5 + (3*a^2*b + b)*x^4 + (a^3 + a)*x^3 + (b^2*x^5 + 2*a*b*x^4 + (a^ 2 + 1)*x^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)), x)
\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^4} \,d x \]