Optimal. Leaf size=121 \[ -\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \text {ArcSin}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))-\frac {i d (a+b \text {ArcSin}(c x))^2}{2 b}+d (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} i b d \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 121, 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 = {4773, 4721,
3798, 2221, 2317, 2438, 201, 222} \begin {gather*} \frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))-\frac {i d (a+b \text {ArcSin}(c x))^2}{2 b}+d \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{4} b d \text {ArcSin}(c x)-\frac {1}{4} b c d x \sqrt {1-c^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4773
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+d \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx-\frac {1}{2} (b c d) \int \sqrt {1-c^2 x^2} \, dx\\ &=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+d \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{4} (b c d) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \sin ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-(2 i d) \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 {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \sin ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-(b d) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \sin ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+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 d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \sin ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+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 d \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 115, normalized size = 0.95 \begin {gather*} -\frac {1}{2} a c^2 d x^2-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}+\frac {1}{4} b d \text {ArcSin}(c x)-\frac {1}{2} b c^2 d x^2 \text {ArcSin}(c x)+b d \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+a d \log (x)-\frac {1}{2} i b 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 = 0.21, size = 164, normalized size = 1.36
method | result | size |
derivativedivides | \(-\frac {d a \,c^{2} x^{2}}{2}+d a \ln \left (c x \right )-\frac {i b d \arcsin \left (c x \right )^{2}}{2}+d b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}-\frac {d b \sin \left (2 \arcsin \left (c x \right )\right )}{8}\) | \(164\) |
default | \(-\frac {d a \,c^{2} x^{2}}{2}+d a \ln \left (c x \right )-\frac {i b d \arcsin \left (c x \right )^{2}}{2}+d b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}-\frac {d b \sin \left (2 \arcsin \left (c x \right )\right )}{8}\) | \(164\) |
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}\right )\, dx + \int a c^{2} x\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int b c^{2} x \operatorname {asin}{\left (c x \right )}\, 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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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