Optimal. Leaf size=61 \[ -\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5799, 5569,
4267, 2317, 2438} \begin {gather*} -\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5799
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx &=\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(61)=122\).
time = 0.07, size = 207, normalized size = 3.39 \begin {gather*} -\frac {a \sinh ^{-1}(c x)}{d}-\frac {b \sinh ^{-1}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}-\frac {b \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}+\frac {a \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {a \log \left (1+c^2 x^2\right )}{2 d}-\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.65, size = 71, normalized size = 1.16
method | result | size |
derivativedivides | \(-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a \ln \left (c x \right )}{d}+\frac {b \left (\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}}\right )-\frac {\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}}\right )}{4}\right )}{d}\) | \(71\) |
default | \(-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a \ln \left (c x \right )}{d}+\frac {b \left (\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}}\right )-\frac {\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}}\right )}{4}\right )}{d}\) | \(71\) |
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}{c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{3} + x}\, 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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\left (d\,c^2\,x^2+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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