Optimal. Leaf size=235 \[ \frac{a b^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac{a b^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}+\frac{b^2 \log (x)}{a^2+1}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.485757, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {5865, 5801, 5831, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{a b^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac{a b^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}+\frac{b^2 \log (x)}{a^2+1}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5801
Rule 5831
Rule 3324
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a+b x)^2}{x^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)^2}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2 \sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\operatorname{Subst}\left (\int \frac{x}{\left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac{b \operatorname{Subst}\left (\int \frac{\cosh (x)}{-\frac{a}{b}+\frac{\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{1+a^2}-\frac{(a b) \operatorname{Subst}\left (\int \frac{x}{-\frac{a}{b}+\frac{\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{1+a^2}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{1}{b}-\frac{2 a e^x}{b}+\frac{e^{2 x}}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{1+a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+x} \, dx,x,\frac{a}{b}+x\right )}{1+a^2}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \log (x)}{1+a^2}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}+\frac{2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{\left (1+a^2\right )^{3/2}}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}+\frac{2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{\left (1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1+a^2}+\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{\left (1+a^2\right )^{3/2}}-\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{\left (1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1+a^2}+\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{\left (1+a^2\right )^{3/2}}-\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{\left (1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1+a^2}+\frac{a b^2 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac{a b^2 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.123856, size = 279, normalized size = 1.19 \[ -\frac{-2 a b^2 x^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )+2 a b^2 x^2 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )-2 \sqrt{a^2+1} b^2 x^2 \log (x)+2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right )-2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right )+2 \sqrt{a^2+1} b x \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)+a^2 \sqrt{a^2+1} \sinh ^{-1}(a+b x)^2+\sqrt{a^2+1} \sinh ^{-1}(a+b x)^2}{2 \left (a^2+1\right )^{3/2} x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.224, size = 384, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{\it Arcsinh} \left ( bx+a \right ) }{{a}^{2}+1}}-{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{ \left ( 2\,{a}^{2}+2 \right ){x}^{2}}}-{\frac{b{\it Arcsinh} \left ( bx+a \right ) }{ \left ({a}^{2}+1 \right ) x}\sqrt{1+ \left ( bx+a \right ) ^{2}}}-{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}{ \left ( 2\,{a}^{2}+2 \right ){x}^{2}}}-{{b}^{2}a{\it Arcsinh} \left ( bx+a \right ) \ln \left ({ \left ( \sqrt{{a}^{2}+1}-bx-\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) \left ( a+\sqrt{{a}^{2}+1} \right ) ^{-1}} \right ) \left ({a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{{b}^{2}a{\it Arcsinh} \left ( bx+a \right ) \ln \left ({ \left ( \sqrt{{a}^{2}+1}+bx+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) \left ( -a+\sqrt{{a}^{2}+1} \right ) ^{-1}} \right ) \left ({a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{{b}^{2}a{\it dilog} \left ({ \left ( \sqrt{{a}^{2}+1}-bx-\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) \left ( a+\sqrt{{a}^{2}+1} \right ) ^{-1}} \right ) \left ({a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{{b}^{2}a{\it dilog} \left ({ \left ( \sqrt{{a}^{2}+1}+bx+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) \left ( -a+\sqrt{{a}^{2}+1} \right ) ^{-1}} \right ) \left ({a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{b}^{2}\ln \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) }{{a}^{2}+1}}+{\frac{{b}^{2}}{{a}^{2}+1}\ln \left ( \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) ^{2}-2\,a \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arsinh}\left (b x + a\right )^{2}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (b x + a\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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