Optimal. Leaf size=235 \[ \frac {a b^2 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a b^2 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}+\frac {b^2 \log (x)}{a^2+1}+\frac {a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{3/2}}-\frac {b \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right ) x}-\frac {\sinh ^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.49, antiderivative size = 235, normalized size of antiderivative = 1.00, 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 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rule 5801
Rule 5831
Rule 5865
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*}
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Mathematica [A] time = 0.13, size = 279, normalized size = 1.19 \[ -\frac {-2 a b^2 x^2 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )+2 a b^2 x^2 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )-2 \sqrt {a^2+1} b^2 x^2 \log (x)+2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac {\sqrt {a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt {a^2+1}+a}\right )-2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac {\sqrt {a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt {a^2+1}-a}\right )+2 \sqrt {a^2+1} b x \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)+a^2 \sqrt {a^2+1} \sinh ^{-1}(a+b x)^2+\sqrt {a^2+1} \sinh ^{-1}(a+b x)^2}{2 \left (a^2+1\right )^{3/2} x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 384, normalized size = 1.63 \[ \frac {b^{2} \arcsinh \left (b x +a \right )}{a^{2}+1}-\frac {\arcsinh \left (b x +a \right )^{2} a^{2}}{2 x^{2} \left (a^{2}+1\right )}-\frac {b \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{\left (a^{2}+1\right ) x}-\frac {\arcsinh \left (b x +a \right )^{2}}{2 x^{2} \left (a^{2}+1\right )}-\frac {b^{2} a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {b^{2} a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {b^{2} \dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {b^{2} \dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {2 b^{2} \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}+\frac {b^{2} \ln \left (\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}-2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-1\right )}{a^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{2 \, x^{2}} + \int \frac {{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x + a b\right )} + b\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b + b\right )} x^{3} + {\left (a^{3} + a\right )} x^{2} + {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} + 1\right )} x^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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