Optimal. Leaf size=337 \[ -\frac {3 \text {ArcSin}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \text {ArcSin}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \text {ArcSin}(a x) \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {i \text {ArcSin}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {ArcSin}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(a x)}\right )}{2 a c^2}-\frac {3 i \text {PolyLog}\left (2,i e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {3 i \text {ArcSin}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcSin}(a x)}\right )}{2 a c^2}-\frac {3 \text {ArcSin}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 \text {ArcSin}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {3 i \text {PolyLog}\left (4,-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {PolyLog}\left (4,i e^{i \text {ArcSin}(a x)}\right )}{a c^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4747, 4749,
4266, 2611, 6744, 2320, 6724, 4767, 2317, 2438} \begin {gather*} \frac {x \text {ArcSin}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {3 \text {ArcSin}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}-\frac {i \text {ArcSin}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {6 i \text {ArcSin}(a x) \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {ArcSin}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcSin}(a x)}\right )}{2 a c^2}-\frac {3 i \text {ArcSin}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcSin}(a x)}\right )}{2 a c^2}-\frac {3 \text {ArcSin}(a x) \text {Li}_3\left (-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 \text {ArcSin}(a x) \text {Li}_3\left (i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {Li}_2\left (-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {3 i \text {Li}_2\left (i e^{i \text {ArcSin}(a x)}\right )}{a c^2}-\frac {3 i \text {Li}_4\left (-i e^{i \text {ArcSin}(a x)}\right )}{a c^2}+\frac {3 i \text {Li}_4\left (i e^{i \text {ArcSin}(a x)}\right )}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4747
Rule 4749
Rule 4767
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {(3 a) \int \frac {x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{2 c^2}+\frac {\int \frac {\sin ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \int \frac {\sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{c^2}+\frac {\text {Subst}\left (\int x^3 \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{2 a c^2}+\frac {3 \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{2 a c^2}+\frac {3 \text {Subst}\left (\int x \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {(3 i) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}+\frac {(3 i) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {(3 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {(3 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 i \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac {3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 i \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {3 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}-\frac {3 i \text {Li}_4\left (-i e^{i \sin ^{-1}(a x)}\right )}{a c^2}+\frac {3 i \text {Li}_4\left (i e^{i \sin ^{-1}(a x)}\right )}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 234, normalized size = 0.69 \begin {gather*} \frac {-\frac {3 \text {ArcSin}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {a x \text {ArcSin}(a x)^3}{1-a^2 x^2}-12 i \text {ArcSin}(a x) \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )-2 i \text {ArcSin}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )+3 i \left (2+\text {ArcSin}(a x)^2\right ) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(a x)}\right )-3 i \left (2+\text {ArcSin}(a x)^2\right ) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(a x)}\right )-6 \text {ArcSin}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(a x)}\right )+6 \text {ArcSin}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcSin}(a x)}\right )-6 i \text {PolyLog}\left (4,-i e^{i \text {ArcSin}(a x)}\right )+6 i \text {PolyLog}\left (4,i e^{i \text {ArcSin}(a x)}\right )}{2 a c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 438, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {-\frac {\arcsin \left (a x \right )^{2} \left (a x \arcsin \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 i \arcsin \left (a x \right )^{2} \polylog \left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 \arcsin \left (a x \right ) \polylog \left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \polylog \left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {\arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 i \arcsin \left (a x \right )^{2} \polylog \left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 \arcsin \left (a x \right ) \polylog \left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \polylog \left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \dilog \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \dilog \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}}{a}\) | \(438\) |
default | \(\frac {-\frac {\arcsin \left (a x \right )^{2} \left (a x \arcsin \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 i \arcsin \left (a x \right )^{2} \polylog \left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 \arcsin \left (a x \right ) \polylog \left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \polylog \left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {\arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 i \arcsin \left (a x \right )^{2} \polylog \left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 \arcsin \left (a x \right ) \polylog \left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \polylog \left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \dilog \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \dilog \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}}{a}\) | \(438\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.70, size = 57, normalized size = 0.17 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, x}{a^{2} c^{2} x^{2} - c^{2}} - \frac {\log \left (a x + 1\right )}{a c^{2}} + \frac {\log \left (a x - 1\right )}{a c^{2}}\right )} \arcsin \left (a x\right )^{3} \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 {\operatorname {asin}^{3}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 {{\mathrm {asin}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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