Optimal. Leaf size=115 \[ -d \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}-\frac {1}{2} b d \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b d \text {csch}^{-1}(c x)^2-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
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Rubi [A] time = 0.29, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6304, 14, 5789, 6742, 264, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac {1}{2} b d \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )-d \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 264
Rule 2190
Rule 2279
Rule 2325
Rule 2391
Rule 3716
Rule 5659
Rule 5789
Rule 6304
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \frac {-\frac {e}{2 x^2}+d \log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \left (-\frac {e}{2 x^2 \sqrt {1+\frac {x^2}{c^2}}}+\frac {d \log (x)}{\sqrt {1+\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \operatorname {Subst}\left (\int \frac {\log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b d) \operatorname {Subst}\left (\int \frac {\sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b d) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 b d) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b d) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} (b d) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 93, normalized size = 0.81 \[ \frac {2 a c d \log (x)+a c e x^2+b e x \sqrt {\frac {1}{c^2 x^2}+1}+b c \text {csch}^{-1}(c x) \left (e x^2-2 d \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )+b c d \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-b c d \text {csch}^{-1}(c x)^2}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x}\, dx \]
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 \[ 2 \, b c^{2} d \int \frac {x \log \relax (x)}{2 \, {\left (\sqrt {c^{2} x^{2} + 1} c^{2} x^{2} + c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} + 1\right )}}\,{d x} - \frac {1}{2} \, b e x^{2} \log \relax (c) - \frac {1}{2} \, b e x^{2} \log \relax (x) + \frac {1}{2} \, a e x^{2} - b d \log \relax (c) \log \relax (x) - \frac {1}{2} \, b d \log \relax (x)^{2} - \frac {1}{4} \, {\left (2 \, \log \left (c^{2} x^{2} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-c^{2} x^{2}\right )\right )} b d + a d \log \relax (x) + \frac {1}{2} \, {\left (b e x^{2} + 2 \, b d \log \relax (x)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + \frac {b e {\left (2 \, \sqrt {c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{4 \, c^{2}} + \frac {b e \log \left (c^{2} x^{2} + 1\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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