Optimal. Leaf size=235 \[ -\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right ) x}-\frac {\sinh ^{-1}(a+b x)^2}{2 x^2}+\frac {a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {a b^2 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \]
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Rubi [A]
time = 0.35, 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 = {5859, 5828,
5843, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 5828
Rule 5843
Rule 5859
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)^2}{x^3} \, dx &=\frac {\text {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}+\text {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}+\text {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 \text {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) \text {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) \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.09, size = 279, normalized size = 1.19 \begin {gather*} -\frac {2 \sqrt {1+a^2} b x \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)+\sqrt {1+a^2} \sinh ^{-1}(a+b x)^2+a^2 \sqrt {1+a^2} \sinh ^{-1}(a+b x)^2+2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac {a+\sqrt {1+a^2}-e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left (\frac {-a+\sqrt {1+a^2}+e^{\sinh ^{-1}(a+b x)}}{-a+\sqrt {1+a^2}}\right )-2 \sqrt {1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 a b^2 x^2 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2} x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.21, size = 374, normalized size = 1.59
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\arcsinh \left (b x +a \right ) \left (4 \left (b x +a \right ) a -2 a^{2}-2 \left (b x +a \right )^{2}+\arcsinh \left (b x +a \right )+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+a^{2} \arcsinh \left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {\dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {\dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}\right )\) | \(374\) |
default | \(b^{2} \left (-\frac {\arcsinh \left (b x +a \right ) \left (4 \left (b x +a \right ) a -2 a^{2}-2 \left (b x +a \right )^{2}+\arcsinh \left (b x +a \right )+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+a^{2} \arcsinh \left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {\dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {\dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}\right )\) | \(374\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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