Optimal. Leaf size=178 \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+\frac {b e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}-b d e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+b d e \text {csch}^{-1}(c x)^2-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
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Rubi [A] time = 0.42, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6304, 266, 43, 5789, 12, 6742, 264, 321, 215, 2325, 5659, 3716, 2190, 2279, 2391} \[ -b d e \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+\frac {b e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}+b d e \text {csch}^{-1}(c x)^2-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 215
Rule 264
Rule 266
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 )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^3} \, 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^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 d e \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^2 x^2+4 d e \log (x)}{2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 d e \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^2 x^2+4 d e \log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 d e \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^2 x^2}{\sqrt {1+\frac {x^2}{c^2}}}+\frac {4 d e \log (x)}{\sqrt {1+\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 d e \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 {x^2}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(2 b d e) \operatorname {Subst}\left (\int \frac {\log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {\left (b e^2\right ) \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 c d^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{4} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )-(2 b d e) \operatorname {Subst}\left (\int \frac {\sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {b c d^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(2 b d e) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b c d^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+b d e \text {csch}^{-1}(c x)^2-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(4 b d 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^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+b d e \text {csch}^{-1}(c x)^2-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 b d e) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b c d^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+b d e \text {csch}^{-1}(c x)^2-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b d e) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {b c d^2 \sqrt {1+\frac {1}{c^2 x^2}}}{4 x}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+b d e \text {csch}^{-1}(c x)^2-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-2 d e \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-b d e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.91, size = 187, normalized size = 1.05 \[ \frac {1}{4} \left (-\frac {2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac {b d^2 \left (-c^2 x^2+c^2 x^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )-1\right )}{c x^3 \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 b e^2 x \left (\sqrt {\frac {1}{c^2 x^2}+1}+c x \text {csch}^{-1}(c x)\right )}{c}-\frac {2 b d^2 \text {csch}^{-1}(c x)}{x^2}+4 b d e \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-4 b d e \text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arcsch}\left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{3}}\, dx \]
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 \[ 4 \, b c^{2} d e \int \frac {x \log \relax (x)}{2 \, {\left (\sqrt {c^{2} x^{2} + 1} c^{2} x^{2} + c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} + 1\right )}}\,{d x} - \frac {1}{2} \, b e^{2} x^{2} \log \relax (c) - \frac {1}{2} \, b e^{2} x^{2} \log \relax (x) + \frac {1}{2} \, a e^{2} x^{2} - 2 \, b d e \log \relax (c) \log \relax (x) - b d e \log \relax (x)^{2} - \frac {1}{2} \, {\left (2 \, \log \left (c^{2} x^{2} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-c^{2} x^{2}\right )\right )} b d e + \frac {1}{8} \, b d^{2} {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac {4 \, \operatorname {arcsch}\left (c x\right )}{x^{2}}\right )} + 2 \, a d e \log \relax (x) + \frac {b e^{2} {\left (2 \, \sqrt {c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{4 \, c^{2}} + \frac {b e^{2} \log \left (c^{2} x^{2} + 1\right )}{4 \, c^{2}} + \frac {1}{2} \, {\left (b e^{2} x^{2} + 4 \, b d e \log \relax (x)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) - \frac {a d^{2}}{2 \, x^{2}} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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