Optimal. Leaf size=73 \[ -\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^2 d}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^2 d}+\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^2 d} \]
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Rubi [A]
time = 0.08, 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 = {5797, 3799,
2221, 2317, 2438} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5797
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx &=\frac {\text {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 \text {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 \text {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 \text {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] Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(73)=146\).
time = 0.05, size = 167, normalized size = 2.29 \begin {gather*} -\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 (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d}+\frac {a \log \left (1+c^2 x^2\right )}{2 c^2 d}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.78, size = 90, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {b \arcsinh \left (c x \right )^{2}}{2 d}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}}{c^{2}}\) | \(90\) |
default | \(\frac {\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {b \arcsinh \left (c x \right )^{2}}{2 d}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}}{c^{2}}\) | \(90\) |
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*} \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} \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 = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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