Optimal. Leaf size=178 \[ \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {1}{4} d (a+b \text {ArcSin}(c x))^2+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {i d (a+b \text {ArcSin}(c x))^3}{3 b}+d (a+b \text {ArcSin}(c x))^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-i b d (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{2} b^2 d \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4787, 4721,
3798, 2221, 2611, 2320, 6724, 4741, 4737, 30} \begin {gather*} \frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-i b d \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {i d (a+b \text {ArcSin}(c x))^3}{3 b}-\frac {1}{4} d (a+b \text {ArcSin}(c x))^2+d \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2+\frac {1}{2} b^2 d \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{4} b^2 c^2 d x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 4737
Rule 4741
Rule 4787
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-(b c d) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+d \text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{2} (b c d) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{2} \left (b^2 c^2 d\right ) \int x \, dx\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-(2 i d) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-(2 b d) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^2 d\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \left (b^2 d\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} b^2 d \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 256, normalized size = 1.44 \begin {gather*} \frac {1}{2} d \left (-a^2 c^2 x^2-2 a b c^2 x^2 \text {ArcSin}(c x)-a b \left (c x \sqrt {1-c^2 x^2}-2 \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{4} b^2 \left (-1+2 \text {ArcSin}(c x)^2\right ) \cos (2 \text {ArcSin}(c x))+4 a b \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+2 a^2 \log (x)-2 i a b \left (\text {ArcSin}(c x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+\frac {1}{12} b^2 \left (-i \pi ^3+8 i \text {ArcSin}(c x)^3+24 \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+24 i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )+12 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )\right )-\frac {1}{2} b^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 420 vs. \(2 (192 ) = 384\).
time = 0.28, size = 421, normalized size = 2.37
method | result | size |
derivativedivides | \(-\frac {d \,a^{2} c^{2} x^{2}}{2}+d \,a^{2} \ln \left (c x \right )-\frac {i d \,b^{2} \arcsin \left (c x \right )^{3}}{3}+d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {d \,b^{2} \arcsin \left (c x \right )^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{4}-\frac {d \,b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {d \,b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{4}-2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d a b \arcsin \left (c x \right )^{2}+\frac {d a b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{2}-\frac {d a b \sin \left (2 \arcsin \left (c x \right )\right )}{4}\) | \(421\) |
default | \(-\frac {d \,a^{2} c^{2} x^{2}}{2}+d \,a^{2} \ln \left (c x \right )-\frac {i d \,b^{2} \arcsin \left (c x \right )^{3}}{3}+d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {d \,b^{2} \arcsin \left (c x \right )^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{4}-\frac {d \,b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {d \,b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{4}-2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d a b \arcsin \left (c x \right )^{2}+\frac {d a b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{2}-\frac {d a b \sin \left (2 \arcsin \left (c x \right )\right )}{4}\) | \(421\) |
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^{2}}{x}\right )\, dx + \int a^{2} c^{2} x\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int b^{2} c^{2} x \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a 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 )}^2\,\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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