Optimal. Leaf size=82 \[ \frac {i (a+b \text {ArcSin}(c x))^2}{2 b c^2 d}-\frac {(a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c^2 d}+\frac {i b \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{2 c^2 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4765, 3800,
2221, 2317, 2438} \begin {gather*} \frac {i (a+b \text {ArcSin}(c x))^2}{2 b c^2 d}-\frac {\log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^2 d}+\frac {i b \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4765
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=\frac {\text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d}\\ &=\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}\\ &=\frac {i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac {\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}+\frac {i b \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(244\) vs. \(2(82)=164\).
time = 0.04, size = 244, normalized size = 2.98 \begin {gather*} -\frac {2 i b \pi \text {ArcSin}(c x)-i b \text {ArcSin}(c x)^2+4 b \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+2 b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+2 b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+a \log \left (1-c^2 x^2\right )-4 b \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-2 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-2 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{2 c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 107, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a \ln \left (c x +1\right )}{2 d}+\frac {i b \arcsin \left (c x \right )^{2}}{2 d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}}{c^{2}}\) | \(107\) |
default | \(\frac {-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a \ln \left (c x +1\right )}{2 d}+\frac {i b \arcsin \left (c x \right )^{2}}{2 d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}}{c^{2}}\) | \(107\) |
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*} - \frac {\int \frac {a x}{c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \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 {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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