Optimal. Leaf size=178 \[ -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e^2 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (6 c^2 d-e\right )}{6 c^3}-\frac {1}{2} b d^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
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Rubi [A] time = 0.42, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {6304, 266, 43, 5789, 6742, 453, 264, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac {1}{2} b d^2 \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (6 c^2 d-e\right )}{6 c^3}+\frac {b e^2 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 264
Rule 266
Rule 453
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 )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \frac {-\frac {e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \left (-\frac {e \left (e+4 d x^2\right )}{4 x^4 \sqrt {1+\frac {x^2}{c^2}}}+\frac {d^2 \log (x)}{\sqrt {1+\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2\right ) \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 {e+4 d x^2}{x^4 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )-\frac {\left (b \left (6 c^2 d-e\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (2 b d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (b d^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} \left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.42, size = 148, normalized size = 0.83 \[ a d^2 \log (x)+a d e x^2+\frac {1}{4} a e^2 x^4+\frac {b d e x \left (\sqrt {\frac {1}{c^2 x^2}+1}+c x \text {csch}^{-1}(c x)\right )}{c}+\frac {b e^2 x \left (3 c^3 x^3 \text {csch}^{-1}(c x)+\sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 x^2-2\right )\right )}{12 c^3}+\frac {1}{2} b d^2 \left (\text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-\text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arcsch}\left (c x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x}\, dx \]
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 \[ \frac {1}{4} \, a e^{2} x^{4} + 4 \, b c^{2} d^{2} \int \frac {x \log \relax (x)}{4 \, {\left (\sqrt {c^{2} x^{2} + 1} c^{2} x^{2} + c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} + 1\right )}}\,{d x} + a d e x^{2} - b d^{2} \log \relax (c) \log \relax (x) - \frac {1}{4} \, {\left (2 \, \log \left (c^{2} x^{2} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-c^{2} x^{2}\right )\right )} b d^{2} + a d^{2} \log \relax (x) + \frac {b d e {\left (2 \, \sqrt {c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{2 \, c^{2}} - \frac {{\left (3 \, c^{2} x^{2} - 2 \, {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {c^{2} x^{2} + 1} - 3 \, \log \left (c^{2} x^{2} + 1\right ) + 3\right )} b e^{2}}{24 \, c^{4}} - \frac {2 \, b c^{2} e^{2} x^{4} \log \relax (c) + 4 \, b c^{2} d^{2} \log \relax (x)^{2} + {\left (8 \, c^{2} d e \log \relax (c) - e^{2}\right )} b x^{2} + 2 \, {\left (b c^{2} e^{2} x^{4} + 4 \, b c^{2} d e x^{2}\right )} \log \relax (x) - 2 \, {\left (b c^{2} e^{2} x^{4} + 4 \, b c^{2} d e x^{2} + 4 \, b c^{2} d^{2} \log \relax (x)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{8 \, c^{2}} + \frac {{\left (4 \, c^{2} d e - e^{2}\right )} b \log \left (c^{2} x^{2} + 1\right )}{8 \, c^{4}} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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