Optimal. Leaf size=210 \[ \frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c^3 d}+\frac {(a+b \text {ArcSin}(c x))^2}{4 c^4 d}-\frac {x^2 (a+b \text {ArcSin}(c x))^2}{2 c^2 d}+\frac {i (a+b \text {ArcSin}(c x))^3}{3 b c^4 d}-\frac {(a+b \text {ArcSin}(c x))^2 \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c^4 d}+\frac {i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c^4 d}-\frac {b^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{2 c^4 d} \]
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Rubi [A]
time = 0.27, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4795, 4765,
3800, 2221, 2611, 2320, 6724, 4737, 30} \begin {gather*} \frac {i b \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^4 d}+\frac {i (a+b \text {ArcSin}(c x))^3}{3 b c^4 d}+\frac {(a+b \text {ArcSin}(c x))^2}{4 c^4 d}-\frac {\log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{c^4 d}-\frac {x^2 (a+b \text {ArcSin}(c x))^2}{2 c^2 d}-\frac {b x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c^3 d}-\frac {b^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 c^4 d}+\frac {b^2 x^2}{4 c^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4737
Rule 4765
Rule 4795
Rule 6724
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 {\text {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) \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 )}{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) \text {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 ) \text {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 \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(459\) vs. \(2(210)=420\).
time = 0.25, size = 459, normalized size = 2.19 \begin {gather*} -\frac {12 a^2 c^2 x^2+12 a b c x \sqrt {1-c^2 x^2}+48 i a b \pi \text {ArcSin}(c x)+24 a b c^2 x^2 \text {ArcSin}(c x)-24 i a b \text {ArcSin}(c x)^2-8 i b^2 \text {ArcSin}(c x)^3-24 a b \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+3 b^2 \cos (2 \text {ArcSin}(c x))-6 b^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))+96 a b \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+24 a b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+48 a b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-24 a b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+48 a b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+24 b^2 \text {ArcSin}(c x)^2 \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )+12 a^2 \log \left (1-c^2 x^2\right )-96 a b \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+24 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-24 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-48 i a b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-48 i a b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )-24 i b^2 \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+12 b^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )+6 b^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))}{24 c^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 380, normalized size = 1.81
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} c^{2} x^{2}}{2 d}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}-\frac {a^{2} \ln \left (c x +1\right )}{2 d}+\frac {i a b \arcsin \left (c x \right )^{2}}{d}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{2 d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}-\frac {b^{2}}{8 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 d}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {a b \arcsin \left (c x \right ) c^{2} x^{2}}{d}+\frac {a b \arcsin \left (c x \right )}{2 d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\) | \(380\) |
default | \(\frac {-\frac {a^{2} c^{2} x^{2}}{2 d}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}-\frac {a^{2} \ln \left (c x +1\right )}{2 d}+\frac {i a b \arcsin \left (c x \right )^{2}}{d}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{2 d}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d}+\frac {b^{2} c^{2} x^{2}}{4 d}-\frac {b^{2}}{8 d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i b^{2} \arcsin \left (c x \right )^{3}}{3 d}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, c x}{2 d}-\frac {a b \arcsin \left (c x \right ) c^{2} x^{2}}{d}+\frac {a b \arcsin \left (c x \right )}{2 d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\) | \(380\) |
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^{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} \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^3\,{\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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