Optimal. Leaf size=128 \[ \frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b e \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {6439, 14,
5822, 12, 6874, 327, 221, 2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)-\frac {1}{2} b e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 221
Rule 327
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^3} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {d x^2+2 e \log (x)}{2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {d x^2+2 e \log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (\frac {d x^2}{\sqrt {1+\frac {x^2}{c^2}}}+\frac {2 e \log (x)}{\sqrt {1+\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(b e) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{4} (b c d) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )-(b e) \text {Subst}\left (\int \frac {\sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b e) \text {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 b e) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b e) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} (b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 138, normalized size = 1.08 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d}{x^2}-\frac {2 b d \text {csch}^{-1}(c x)}{x^2}-\frac {b d \left (-1-c^2 x^2+c^2 x^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\right )}{c \sqrt {1+\frac {1}{c^2 x^2}} x^3}-2 b e \text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )+4 a e \log (x)+2 b e \text {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{3}}\, dx\]
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^{3}}\, 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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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