Optimal. Leaf size=86 \[ -\frac {\text {Li}_2\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {(a+b x)^2+1}}+\frac {\sinh ^{-1}(a+b x)^2}{b}-\frac {2 \sinh ^{-1}(a+b x) \log \left (e^{2 \sinh ^{-1}(a+b x)}+1\right )}{b} \]
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Rubi [A] time = 0.16, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5867, 5687, 5714, 3718, 2190, 2279, 2391} \[ -\frac {\text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {(a+b x)^2+1}}+\frac {\sinh ^{-1}(a+b x)^2}{b}-\frac {2 \sinh ^{-1}(a+b x) \log \left (e^{2 \sinh ^{-1}(a+b x)}+1\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5687
Rule 5714
Rule 5867
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)^2}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\left (1+x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \operatorname {Subst}\left (\int \frac {x \sinh ^{-1}(x)}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {4 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \sinh ^{-1}(a+b x) \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac {2 \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \sinh ^{-1}(a+b x) \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \sinh ^{-1}(a+b x) \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}-\frac {\text {Li}_2\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 98, normalized size = 1.14 \[ \frac {\sinh ^{-1}(a+b x) \left (\frac {\left (-\sqrt {a^2+2 a b x+b^2 x^2+1}+a+b x\right ) \sinh ^{-1}(a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2+1}}-2 \log \left (e^{-2 \sinh ^{-1}(a+b x)}+1\right )\right )+\text {Li}_2\left (-e^{-2 \sinh ^{-1}(a+b x)}\right )}{b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 168, normalized size = 1.95 \[ -\frac {\left (b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b +2 a b x -\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +a^{2}+1\right ) \arcsinh \left (b x +a \right )^{2}}{b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}+\frac {2 \arcsinh \left (b x +a \right )^{2}}{b}-\frac {2 \arcsinh \left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}-\frac {\polylog \left (2, -\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b} \]
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 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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