Optimal. Leaf size=118 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x} \]
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Rubi [A] time = 0.19727, antiderivative size = 118, 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, 95, 5721, 5461, 4182, 2279, 2391} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x} \]
Antiderivative was successfully verified.
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Rule 5746
Rule 95
Rule 5721
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+c^2 \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx+\frac{(b c) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}-\frac{c^2 \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d}+\frac{b c^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{b c^2 \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.527178, size = 144, normalized size = 1.22 \[ -\frac{b c^2 \left (\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\frac{\cosh ^{-1}(c x)}{c^2 x^2}-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+a c^2 \log \left (1-c^2 x^2\right )-2 a c^2 \log (x)+\frac{a}{x^2}}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.092, size = 301, normalized size = 2.6 \begin{align*} -{\frac{{c}^{2}a\ln \left ( cx-1 \right ) }{2\,d}}-{\frac{a}{2\,d{x}^{2}}}+{\frac{{c}^{2}a\ln \left ( cx \right ) }{d}}-{\frac{{c}^{2}a\ln \left ( cx+1 \right ) }{2\,d}}+{\frac{bc}{2\,dx}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{c}^{2}b}{2\,d}}-{\frac{b{\rm arccosh} \left (cx\right )}{2\,d{x}^{2}}}+{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) }+{\frac{{c}^{2}b}{2\,d}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b}{d}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b}{d}{\it polylog} \left ( 2,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}{2} \,{\left (\frac{c^{2} \log \left (c x + 1\right )}{d} + \frac{c^{2} \log \left (c x - 1\right )}{d} - \frac{2 \, c^{2} \log \left (x\right )}{d} + \frac{1}{d x^{2}}\right )} a - b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{2} d x^{5} - d x^{3}}\,{d x} \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^{5} - d x^{3}}, 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^{5} - x^{3}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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