Optimal. Leaf size=139 \[ -\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \text {ArcSin}(c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {i c^2 d (a+b \text {ArcSin}(c x))^2}{2 b}-c^2 d (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{2} i b c^2 d \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4775, 283, 222,
4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {i c^2 d (a+b \text {ArcSin}(c x))^2}{2 b}-c^2 d \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+\frac {1}{2} i b c^2 d \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} b c^2 d \text {ArcSin}(c x)-\frac {b c d \sqrt {1-c^2 x^2}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4775
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} (b c d) \int \frac {\sqrt {1-c^2 x^2}}{x^2} \, dx-\left (c^2 d\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (c^2 d\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{2} \left (b c^3 d\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \sin ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (2 i c^2 d\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \sin ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\left (b c^2 d\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \sin ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} \left (i b c^2 d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \sin ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} i b c^2 d \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 112, normalized size = 0.81 \begin {gather*} -\frac {a d}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {b d \text {ArcSin}(c x)}{2 x^2}-b c^2 d \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-a c^2 d \log (x)+\frac {1}{2} i b c^2 d \left (\text {ArcSin}(c x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 186, normalized size = 1.34
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {d a}{2 c^{2} x^{2}}-d a \ln \left (c x \right )+\frac {i b d \arcsin \left (c x \right )^{2}}{2}+\frac {i d b}{2}-\frac {d b \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {d b \arcsin \left (c x \right )}{2 c^{2} x^{2}}-d b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-d b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(186\) |
default | \(c^{2} \left (-\frac {d a}{2 c^{2} x^{2}}-d a \ln \left (c x \right )+\frac {i b d \arcsin \left (c x \right )^{2}}{2}+\frac {i d b}{2}-\frac {d b \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {d b \arcsin \left (c x \right )}{2 c^{2} x^{2}}-d b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-d b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(186\) |
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*} - d \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {a c^{2}}{x}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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