Optimal. Leaf size=61 \[ -\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \]
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Rubi [A] time = 0.12, 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 = {5720, 5461, 4182, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 5461
Rule 5720
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=\frac {2 \operatorname {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 \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {b \operatorname {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 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {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] time = 0.08, size = 207, normalized size = 3.39 \[ -\frac {a \log \left (c^2 x^2+1\right )}{2 d}-\frac {a \sinh ^{-1}(c x)}{d}+\frac {a \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b \text {Li}_2\left (-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}-\frac {b \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}-\frac {b \sinh ^{-1}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )}{d}-\frac {b \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )}{d}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{c^{2} d x^{3} + d x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 74, normalized size = 1.21 \[ \frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b \dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}}\right )}{d}-\frac {b \dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}}\right )}{4 d} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{d} - \frac {2 \, \log \relax (x)}{d}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d x^{3} + d x}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\left (d\,c^2\,x^2+d\right )} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{3} + x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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