Optimal. Leaf size=113 \[ -\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b c^2 \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {b c^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 113, 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 = {5809, 5799,
5569, 4267, 2317, 2438, 270} \begin {gather*} \frac {2 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac {b c^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {b c^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {b c \sqrt {c^2 x^2+1}}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5799
Rule 5809
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}-c^2 \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {1}{x^2 \sqrt {1+c^2 x^2}} \, dx}{2 d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}-\frac {c^2 \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b c^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {b c^2 \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. \(240\) vs. \(2(113)=226\).
time = 0.18, size = 240, normalized size = 2.12 \begin {gather*} \frac {-\frac {b c \sqrt {1+c^2 x^2}}{x}-b c^2 \sinh ^{-1}(c x)^2-\frac {a+b \sinh ^{-1}(c x)}{x^2}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{b}+2 b c^2 \sinh ^{-1}(c x) \log \left (1+\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+a c^2 \log \left (1+c^2 x^2\right )+2 b c^2 \text {PolyLog}\left (2,\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \text {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-c^2 \left (2 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.96, size = 251, normalized size = 2.22
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a \ln \left (c x \right )}{d}-\frac {b \sqrt {c^{2} x^{2}+1}}{2 d c x}+\frac {b}{2 d}-\frac {b \arcsinh \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}-\frac {b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}\right )\) | \(251\) |
default | \(c^{2} \left (\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a \ln \left (c x \right )}{d}-\frac {b \sqrt {c^{2} x^{2}+1}}{2 d c x}+\frac {b}{2 d}-\frac {b \arcsinh \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}-\frac {b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}\right )\) | \(251\) |
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^{5} + x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, 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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,\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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