Optimal. Leaf size=218 \[ \frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^3 d}-\frac {x (a+b \text {ArcSin}(c x))^2}{c^2 d}-\frac {2 i (a+b \text {ArcSin}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^3 d}-\frac {2 b^2 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d}+\frac {2 b^2 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{c^3 d} \]
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Rubi [A]
time = 0.20, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4795, 4749,
4266, 2611, 2320, 6724, 4767, 8} \begin {gather*} -\frac {2 i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{c^3 d}+\frac {2 i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d}-\frac {2 i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d}-\frac {x (a+b \text {ArcSin}(c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^3 d}-\frac {2 b^2 \text {Li}_3\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^3 d}+\frac {2 b^2 \text {Li}_3\left (i e^{i \text {ArcSin}(c x)}\right )}{c^3 d}+\frac {2 b^2 x}{c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2320
Rule 2611
Rule 4266
Rule 4749
Rule 4767
Rule 4795
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx &=-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{c^2}+\frac {(2 b) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{c d}\\ &=-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac {\text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d}+\frac {\left (2 b^2\right ) \int 1 \, dx}{c^2 d}\\ &=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d}+\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d}+\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^3 d}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^3 d}\\ &=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^3 d}+\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}-\frac {2 b^2 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}+\frac {2 b^2 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{c^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 317, normalized size = 1.45 \begin {gather*} -\frac {2 a^2 c x-4 b^2 c x+4 a b \sqrt {1-c^2 x^2}+4 a b c x \text {ArcSin}(c x)+4 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+2 b^2 c x \text {ArcSin}(c x)^2-4 a b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-2 b^2 \text {ArcSin}(c x)^2 \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+4 a b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+2 b^2 \text {ArcSin}(c x)^2 \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+a^2 \log (1-c x)-a^2 \log (1+c x)-4 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+4 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )+4 b^2 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )-4 b^2 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{2 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{-c^{2} d \,x^{2}+d}\, dx\]
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^{2} x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{2} \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 = -0.00 \begin {gather*} \int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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