Optimal. Leaf size=95 \[ \frac{b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{2 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d} \]
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Rubi [A] time = 0.139802, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {5746, 92, 205, 5694, 4182, 2279, 2391} \[ \frac{b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{2 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5746
Rule 92
Rule 205
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{d x}+c^2 \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx+\frac{(b c) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{d x}-\frac{c \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{b c \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}+\frac{2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{(b c) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac{(b c) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{b c \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}+\frac{2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{(b c) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{b c \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}+\frac{2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{b c \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.358219, size = 132, normalized size = 1.39 \[ \frac{b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{a+b \cosh ^{-1}(c x)}{x}-c \log \left (1-e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )+c \log \left (e^{\cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{b c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}}{d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.108, size = 161, normalized size = 1.7 \begin{align*} -{\frac{a}{dx}}-{\frac{ca\ln \left ( cx-1 \right ) }{2\,d}}+{\frac{ca\ln \left ( cx+1 \right ) }{2\,d}}-{\frac{b{\rm arccosh} \left (cx\right )}{dx}}+2\,{\frac{bc\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{d}}+{\frac{bc}{d}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{bc}{d}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{bc{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \,{\left (24 \, c^{3} \int \frac{x \log \left (c x - 1\right )}{4 \,{\left (c^{2} d x^{2} - d\right )}}\,{d x} - 4 \, c^{2}{\left (\frac{\log \left (c x + 1\right )}{c d} - \frac{\log \left (c x - 1\right )}{c d}\right )} - 8 \, c^{2} \int \frac{\log \left (c x - 1\right )}{4 \,{\left (c^{2} d x^{2} - d\right )}}\,{d x} - \frac{c x \log \left (c x + 1\right )^{2} + 2 \, c x \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \,{\left (c x \log \left (c x + 1\right ) - c x \log \left (c x - 1\right ) - 2\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{d x} + 8 \, \int \frac{c^{2} x \log \left (c x + 1\right ) - c^{2} x \log \left (c x - 1\right ) - 2 \, c}{2 \,{\left (c^{3} d x^{4} - c d x^{2} +{\left (c^{2} d x^{3} - d x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}}\,{d x}\right )} b + \frac{1}{2} \, a{\left (\frac{c \log \left (c x + 1\right )}{d} - \frac{c \log \left (c x - 1\right )}{d} - \frac{2}{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{arcosh}\left (c x\right ) + a}{c^{2} d x^{4} - d x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{2} x^{4} - x^{2}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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