Optimal. Leaf size=110 \[ -\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5811, 5799,
5569, 4267, 2317, 2438, 197} \begin {gather*} \frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (c^2 x^2+1\right )}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {b c x}{2 d^2 \sqrt {c^2 x^2+1}}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5799
Rule 5811
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx &=\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {2 \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(110)=220\).
time = 0.30, size = 234, normalized size = 2.13 \begin {gather*} -\frac {\frac {a^2}{b}-\frac {a}{1+c^2 x^2}+\frac {b c x}{\sqrt {1+c^2 x^2}}+2 a \sinh ^{-1}(c x)-\frac {b \sinh ^{-1}(c x)}{1+c^2 x^2}+2 b \sinh ^{-1}(c x) \log \left (1+\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+2 b \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-2 a \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-2 b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+a \log \left (1+c^2 x^2\right )+2 b \text {PolyLog}\left (2,\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+2 b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs.
\(2(127)=254\).
time = 2.87, size = 283, normalized size = 2.57
method | result | size |
derivativedivides | \(-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a \ln \left (c x \right )}{d^{2}}-\frac {b c x}{2 d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}+\frac {b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}\) | \(283\) |
default | \(-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a \ln \left (c x \right )}{d^{2}}-\frac {b c x}{2 d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}+\frac {b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}\) | \(283\) |
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^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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