Optimal. Leaf size=105 \[ \frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}-\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d} \]
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Rubi [A] time = 0.18, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5714, 3718, 2190, 2531, 2282, 6589} \[ \frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}-\frac {b^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 5714
Rule 6589
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 281, normalized size = 2.68 \[ \frac {3 a^2 \log \left (c^2 x^2+1\right )+12 b \text {Li}_2\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right ) \left (a+b \sinh ^{-1}(c x)\right )+12 b \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right ) \left (a+b \sinh ^{-1}(c x)\right )+12 a b \sinh ^{-1}(c x) \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )+12 a b \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-6 a b \sinh ^{-1}(c x)^2-12 b^2 \text {Li}_3\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )-12 b^2 \text {Li}_3\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+6 b^2 \sinh ^{-1}(c x)^2 \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )+6 b^2 \sinh ^{-1}(c x)^2 \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-2 b^2 \sinh ^{-1}(c x)^3}{6 c^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arsinh}\left (c x\right ) + a^{2} x}{c^{2} d x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x}{c^{2} d x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 223, normalized size = 2.12 \[ \frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2} d}-\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 c^{2} d}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{2} d}+\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{2} d}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{2} d}-\frac {a b \arcsinh \left (c x \right )^{2}}{c^{2} d}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{2} d}+\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{2} d} \]
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 \[ \frac {b^{2} \log \left (c^{2} x^{2} + 1\right ) \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{2 \, c^{2} d} + \frac {a^{2} \log \left (c^{2} d x^{2} + d\right )}{2 \, c^{2} d} - \int -\frac {{\left (2 \, a b c^{2} x^{2} - {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) - {\left (b^{2} c x \log \left (c^{2} x^{2} + 1\right ) - 2 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d x^{3} + c^{2} d x + {\left (c^{3} d x^{2} + c d\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \]
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 \[ \frac {\int \frac {a^{2} x}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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