Optimal. Leaf size=124 \[ -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )+b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} i b d \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5348, 14,
4816, 6874, 270, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -d \log \left (\frac {1}{x}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}-\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} i b d \csc ^{-1}(c x)^2+b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2221
Rule 2317
Rule 2363
Rule 2438
Rule 3798
Rule 4721
Rule 4816
Rule 5348
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {b \text {Subst}\left (\int \frac {-\frac {e}{2 x^2}+d \log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-d \left (a+b \sec ^{-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^2}{c^2}}}+\frac {d \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {(b d) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}+\frac {(b e) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b d) \text {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b d) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(2 i b d) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )+b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b d) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )+b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} (i b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sec ^{-1}(c x)\right )+b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 104, normalized size = 0.84 \begin {gather*} \frac {-b e \sqrt {1-\frac {1}{c^2 x^2}} x+a c e x^2+i b c d \sec ^{-1}(c x)^2+b c \sec ^{-1}(c x) \left (e x^2-2 d \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )+2 a c d \log (x)+i b c d \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.99, size = 142, normalized size = 1.15
method | result | size |
derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {i b d \mathrm {arcsec}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e \,x^{2}}{2}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e x}{2 c}-\frac {i b e}{2 c^{2}}-b d \,\mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {i b d \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}\) | \(142\) |
default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {i b d \mathrm {arcsec}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e \,x^{2}}{2}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e x}{2 c}-\frac {i b e}{2 c^{2}}-b d \,\mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {i b d \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}\) | \(142\) |
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 {asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = -0.01 \begin {gather*} \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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