Optimal. Leaf size=332 \[ \frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x (a+b \text {ArcSin}(c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {ArcSin}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {ArcSin}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 i b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4791, 4767,
4749, 4266, 2317, 2438, 267} \begin {gather*} -\frac {10 i b \sqrt {1-c^2 x^2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {ArcSin}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {ArcSin}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x (a+b \text {ArcSin}(c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2438
Rule 4266
Rule 4749
Rule 4767
Rule 4791
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 511, normalized size = 1.54 \begin {gather*} \frac {8 a^2-2 b^2-12 a^2 c^2 x^2+4 a b \text {ArcSin}(c x)+2 b^2 \text {ArcSin}(c x)^2-2 b^2 \cos (2 \text {ArcSin}(c x))+12 a b \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))+6 b^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))-15 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-5 b^2 \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+15 b^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+5 b^2 \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x)) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+15 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+5 a b \cos (3 \text {ArcSin}(c x)) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-15 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-5 a b \cos (3 \text {ArcSin}(c x)) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-20 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+20 i b^2 \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )+2 a b \sin (2 \text {ArcSin}(c x))+2 b^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))}{12 c^4 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \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. 827 vs. \(2 (315 ) = 630\).
time = 0.41, size = 828, normalized size = 2.49
method | result | size |
default | \(a^{2} \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} x^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{3}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {5 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{3}}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {5 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{3 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}\) | \(828\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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