Optimal. Leaf size=425 \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 d}+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d} \]
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Rubi [A] time = 0.862468, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6304, 5791, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 d}+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d} \]
Antiderivative was successfully verified.
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Rule 6304
Rule 5791
Rule 5799
Rule 5561
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{-d}}+\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{2 d}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 0.957291, size = 387, normalized size = 0.91 \[ \frac{b \left (\text{PolyLog}\left (2,\frac{\left (2 \sqrt{e \left (e-c^2 d\right )}+c^2 d-2 e\right ) e^{-2 \text{csch}^{-1}(c x)}}{c^2 d}\right )+\text{PolyLog}\left (2,\frac{\left (c^2 d-2 \left (\sqrt{e \left (e-c^2 d\right )}+e\right )\right ) e^{-2 \text{csch}^{-1}(c x)}}{c^2 d}\right )-2 \left (i \sin ^{-1}\left (\sqrt{\frac{e}{c^2 d}}\right ) \left (2 \tanh ^{-1}\left (\frac{\sqrt{e \left (e-c^2 d\right )}}{c e x \sqrt{\frac{1}{c^2 x^2}+1}}\right )-\log \left (\frac{e^{-2 \text{csch}^{-1}(c x)} \left (-2 \sqrt{e \left (e-c^2 d\right )}+c^2 d \left (e^{2 \text{csch}^{-1}(c x)}-1\right )+2 e\right )}{c^2 d}\right )+\log \left (\frac{e^{-2 \text{csch}^{-1}(c x)} \left (2 \left (\sqrt{e \left (e-c^2 d\right )}+e\right )+c^2 d \left (e^{2 \text{csch}^{-1}(c x)}-1\right )\right )}{c^2 d}\right )\right )+\text{csch}^{-1}(c x) \left (\log \left (\frac{e^{-2 \text{csch}^{-1}(c x)} \left (-2 \sqrt{e \left (e-c^2 d\right )}+c^2 d \left (e^{2 \text{csch}^{-1}(c x)}-1\right )+2 e\right )}{c^2 d}\right )+\log \left (\frac{e^{-2 \text{csch}^{-1}(c x)} \left (2 \left (\sqrt{e \left (e-c^2 d\right )}+e\right )+c^2 d \left (e^{2 \text{csch}^{-1}(c x)}-1\right )\right )}{c^2 d}\right )\right )+\text{csch}^{-1}(c x)^2\right )\right )-2 a \log \left (d+e x^2\right )+4 a \log (x)}{4 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.48, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{x \left ( e{x}^{2}+d \right ) }}\, dx \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} \, a{\left (\frac{\log \left (e x^{2} + d\right )}{d} - \frac{2 \, \log \left (x\right )}{d}\right )} + b \int \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + \frac{1}{c x}\right )}{e x^{3} + d x}\,{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{arcsch}\left (c x\right ) + a}{e x^{3} + d x}, 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*} \int \frac{a + b \operatorname{acsch}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \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{arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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