Optimal. Leaf size=73 \[ -\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d}+\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d} \]
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Rubi [A] time = 0.12, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5714, 3718, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5714
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {b \operatorname {Subst}\left (\int \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 )^2}{2 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 167, normalized size = 2.29 \[ \frac {a \log \left (c^2 x^2+1\right )}{2 c^2 d}+\frac {b \text {Li}_2\left (-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d}+\frac {b \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d}-\frac {b \sinh ^{-1}(c x)^2}{2 c^2 d}+\frac {b \sinh ^{-1}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d}+\frac {b \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )}{c^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \operatorname {arsinh}\left (c x\right ) + a x}{c^{2} d x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{c^{2} d x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 98, normalized size = 1.34 \[ \frac {a \ln \left (c^{2} x^{2}+1\right )}{2 c^{2} d}-\frac {b \arcsinh \left (c x \right )^{2}}{2 c^{2} d}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{2} d}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{2} d} \]
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 \[ -\frac {1}{8} \, b {\left (\frac {\log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right ) \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d} + 8 \, \int \frac {\log \left (c^{2} x^{2} + 1\right )}{2 \, {\left (c^{4} d x^{3} + c^{2} d x + {\left (c^{3} d x^{2} + c d\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} + \frac {a \log \left (c^{2} d x^{2} + d\right )}{2 \, c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \]
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 \[ \frac {\int \frac {a x}{c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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