Optimal. Leaf size=128 \[ -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)-\frac {1}{2} b e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
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Rubi [A] time = 0.30, 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 = {6304, 14, 5789, 12, 6742, 321, 215, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac {1}{2} b e \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d \text {csch}^{-1}(c x)+\frac {1}{2} b e \text {csch}^{-1}(c x)^2-b e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 215
Rule 321
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^3} \, dx &=-\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(b e) \operatorname {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) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )-(b e) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 = 0.57, size = 138, normalized size = 1.08 \[ \frac {1}{4} \left (-\frac {2 a d}{x^2}+4 a e \log (x)-\frac {b d \left (-c^2 x^2+c^2 x^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )-1\right )}{c x^3 \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 b d \text {csch}^{-1}(c x)}{x^2}+2 b e \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-2 b e \text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{3}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (4 \, c^{2} \int \frac {x^{2} \log \relax (x)}{c^{2} x^{3} + x}\,{d x} - 2 \, c^{2} \int \frac {x \log \relax (x)}{c^{2} x^{2} + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 1}\,{d x} - {\left (\log \left (c^{2} x^{2} + 1\right ) - 2 \, \log \relax (x)\right )} \log \relax (c) + \log \left (c^{2} x^{2} + 1\right ) \log \relax (c) - 2 \, \log \relax (x) \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + 2 \, \int \frac {\log \relax (x)}{c^{2} x^{3} + x}\,{d x}\right )} b e + \frac {1}{8} \, b d {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac {4 \, \operatorname {arcsch}\left (c x\right )}{x^{2}}\right )} + a e \log \relax (x) - \frac {a d}{2 \, x^{2}} \]
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 \[ \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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