Optimal. Leaf size=309 \[ \frac {b c d \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]
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Rubi [A]
time = 0.53, antiderivative size = 309, normalized size of antiderivative = 1.00, number
of steps used = 14, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules
used = {6438, 14, 5958, 12, 6874, 92, 54, 2365, 2363, 4721, 3798, 2221, 2317, 2438}
\begin {gather*} -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b 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 {b 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 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 54
Rule 92
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 ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {sech}^{-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}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {sech}^{-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}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {sech}^{-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}{c}} \sqrt {1+\frac {x}{c}}}+\frac {2 e \log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \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 {(b 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 {b c d \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text {sech}^{-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}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )+\frac {\left (b 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 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (b 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 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (b 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 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (2 i b 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 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {b 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 {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b 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 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {b 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 {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (i b 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 )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=\frac {b c d \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {b 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 {b 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}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 164, normalized size = 0.53 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d}{x^2}-\frac {2 b d \text {sech}^{-1}(c x)}{x^2}+\frac {b d \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}}-2 b 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 )+4 a e \log (x)+2 b 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 = 0.58, size = 191, normalized size = 0.62
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} e}{2 c^{2}}+\frac {b d \,\mathrm {arcsech}\left (c x \right )}{4}+\frac {b d \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d}{2 c^{2} x^{2}}-\frac {b e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{2}}\right )\) | \(191\) |
default | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} e}{2 c^{2}}+\frac {b d \,\mathrm {arcsech}\left (c x \right )}{4}+\frac {b d \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d}{2 c^{2} x^{2}}-\frac {b e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{2}}\right )\) | \(191\) |
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 )}{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 )\,\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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