Optimal. Leaf size=215 \[ \frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {b \text {Li}_2\left (\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {Li}_2\left (\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )}{2 e} \]
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Rubi [A] time = 0.39, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6289, 2518} \[ \frac {b \text {PolyLog}\left (2,\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {PolyLog}\left (2,\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )}{2 e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{e} \]
Antiderivative was successfully verified.
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Rule 2518
Rule 6289
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{d+e x} \, dx &=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )}{e}+\frac {b \int \frac {\log \left (1-\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c e}+\frac {b \int \frac {\log \left (1-\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c e}-\frac {b \int \frac {\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )}{e}+\frac {b \text {Li}_2\left (\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {Li}_2\left (\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )}{2 e}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 506, normalized size = 2.35 \[ \frac {a \log (d+e x)}{e}+\frac {b \left (8 \text {Li}_2\left (\frac {\left (e-\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )+8 \text {Li}_2\left (\frac {\left (e+\sqrt {c^2 d^2+e^2}\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )+8 \text {csch}^{-1}(c x) \log \left (\frac {\left (\sqrt {c^2 d^2+e^2}-e\right ) e^{\text {csch}^{-1}(c x)}}{c d}+1\right )+8 \text {csch}^{-1}(c x) \log \left (1-\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )+4 i \pi \log \left (\frac {\left (\sqrt {c^2 d^2+e^2}-e\right ) e^{\text {csch}^{-1}(c x)}}{c d}+1\right )+4 i \pi \log \left (1-\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )+16 i \sin ^{-1}\left (\frac {\sqrt {1+\frac {i e}{c d}}}{\sqrt {2}}\right ) \log \left (\frac {\left (\sqrt {c^2 d^2+e^2}-e\right ) e^{\text {csch}^{-1}(c x)}}{c d}+1\right )-16 i \sin ^{-1}\left (\frac {\sqrt {1+\frac {i e}{c d}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {c^2 d^2+e^2}+e\right ) e^{\text {csch}^{-1}(c x)}}{c d}\right )-32 \sin ^{-1}\left (\frac {\sqrt {1+\frac {i e}{c d}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {(e+i c d) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c x)\right )\right )}{\sqrt {c^2 d^2+e^2}}\right )+4 \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-8 \text {csch}^{-1}(c x)^2-4 i \pi \text {csch}^{-1}(c x)-8 \text {csch}^{-1}(c x) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )-4 i \pi \log \left (\frac {d}{x}+e\right )+\pi ^2\right )}{8 e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcsch}\left (c x\right ) + a}{e x + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{e x +d}\, dx \]
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 \[ b \int \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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