Optimal. Leaf size=373 \[ \frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]
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Rubi [A]
time = 0.70, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used =
{6438, 272, 45, 5958, 12, 6874, 97, 92, 54, 2365, 2363, 4721, 3798, 2221, 2317, 2438}
\begin {gather*} -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x)}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b c d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{4 x}-\frac {b e^2 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 54
Rule 92
Rule 97
Rule 272
Rule 2221
Rule 2317
Rule 2363
Rule 2365
Rule 2438
Rule 3798
Rule 4721
Rule 5958
Rule 6438
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e^2}{x^2}+d^2 x^2+4 d e \log (x)}{2 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e^2}{x^2}+d^2 x^2+4 d e \log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-2 d e \left (a+b \text {sech}^{-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}{c}} \sqrt {1+\frac {x}{c}}}+\frac {d^2 x^2}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}}+\frac {4 d e \log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(2 b d e) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{4} \left (b c d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )+\frac {\left (2 b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (2 b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (2 b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (4 i b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (2 b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (i b d e \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 227, normalized size = 0.61 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2-\frac {2 b e^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c^2}-\frac {2 b d^2 \text {sech}^{-1}(c x)}{x^2}+2 b e^2 x^2 \text {sech}^{-1}(c x)+\frac {b d^2 \sqrt {\frac {1-c x}{1+c x}} \left (\sqrt {1-c x} (1+c x)-2 i c^2 x^2 \sqrt {1+c x} \text {ArcTan}\left (c x+i \sqrt {1-c^2 x^2}\right )\right )}{x^2 \sqrt {1-c x}}-4 b d e \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+8 a d e \log (x)+4 b d e \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 279, normalized size = 0.75
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} d e}{c^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b d e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b d e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}\right )\) | \(279\) |
default | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} d e}{c^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b d e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b d e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}\right )\) | \(279\) |
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 {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{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 = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\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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