Optimal. Leaf size=210 \[ \frac {i b \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}-\frac {b^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}+\frac {b^2 x^2}{4 c^2 d} \]
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Rubi [A] time = 0.38, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {4715, 4675, 3719, 2190, 2531, 2282, 6589, 4707, 4641, 30} \[ \frac {i b \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {b^2 \text {PolyLog}\left (3,-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d}+\frac {b^2 x^2}{4 c^2 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3719
Rule 4641
Rule 4675
Rule 4707
Rule 4715
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx &=-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{c d}\\ &=-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\operatorname {Subst}\left (\int (a+b x)^2 \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d}+\frac {b \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d}+\frac {b^2 \int x \, dx}{2 c^2 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}+\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}\\ &=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d}-\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^4 d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^4 d}-\frac {b^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}\\ \end {align*}
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Mathematica [B] time = 0.40, size = 441, normalized size = 2.10 \[ -\frac {12 a^2 c^2 x^2+12 a^2 \log \left (1-c^2 x^2\right )+12 a b c x \sqrt {1-c^2 x^2}+24 a b c^2 x^2 \sin ^{-1}(c x)-48 i a b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )-48 i a b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )-24 i a b \sin ^{-1}(c x)^2-12 a b \sin ^{-1}(c x)+48 i \pi a b \sin ^{-1}(c x)+48 a b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+48 a b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+96 \pi a b \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+24 \pi a b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-24 \pi a b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-24 \pi a b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-96 \pi a b \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+24 \pi a b \log \left (-\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-24 i b^2 \sin ^{-1}(c x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )+12 b^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )-8 i b^2 \sin ^{-1}(c x)^3+6 b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+24 b^2 \sin ^{-1}(c x)^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )-6 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )+3 b^2 \cos \left (2 \sin ^{-1}(c x)\right )}{24 c^4 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} x^{3} \arcsin \left (c x\right )^{2} + 2 \, a b x^{3} \arcsin \left (c x\right ) + a^{2} x^{3}}{c^{2} d x^{2} - d}, x\right ) \]
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 \[ \int -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{c^{2} d x^{2} - d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 416, normalized size = 1.98 \[ -\frac {a^{2} x^{2}}{2 c^{2} d}-\frac {a^{2} \ln \left (c x -1\right )}{2 c^{4} d}-\frac {a^{2} \ln \left (c x +1\right )}{2 c^{4} d}+\frac {i a b \arcsin \left (c x \right )^{2}}{c^{4} d}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x}{2 c^{3} d}-\frac {b^{2} \arcsin \left (c x \right )^{2} x^{2}}{2 c^{2} d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 c^{4} d}+\frac {b^{2} x^{2}}{4 c^{2} d}-\frac {b^{2}}{8 c^{4} d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{4} d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 c^{4} d}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{4} d}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{4} d}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, x}{2 c^{3} d}-\frac {a b \arcsin \left (c x \right ) x^{2}}{c^{2} d}+\frac {a b \arcsin \left (c x \right )}{2 c^{4} d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{4} d}+\frac {i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \]
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 \[ - \frac {\int \frac {a^{2} x^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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