Optimal. Leaf size=115 \[ \frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.20, 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 = {6439, 14,
5822, 6874, 270, 2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -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) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2221
Rule 2317
Rule 2362
Rule 2438
Rule 3797
Rule 5775
Rule 5822
Rule 6439
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx &=-\text {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 \text {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 \text {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) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 \begin {gather*} \frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x+a c e x^2-b c d \text {csch}^{-1}(c x)^2+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 )+2 a c d \log (x)+b c d \text {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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