Optimal. Leaf size=187 \[ -\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \text {ArcSin}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {ArcSin}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^5 d^2}-\frac {3 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{2 c^5 d^2}+\frac {3 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{2 c^5 d^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4791, 4795,
4749, 4266, 2317, 2438, 267, 272, 45} \begin {gather*} \frac {3 i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^5 d^2}+\frac {3 x (a+b \text {ArcSin}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {ArcSin}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{2 c^5 d^2}+\frac {3 i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{2 c^5 d^2}+\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}-\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4266
Rule 4749
Rule 4791
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac {3 \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(3 b) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d^2}-\frac {b \text {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=\frac {3 b \sqrt {1-c^2 x^2}}{2 c^5 d^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}-\frac {b \text {Subst}\left (\int \left (\frac {1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2}\\ &=-\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}+\frac {(3 b) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}-\frac {(3 b) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=-\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=-\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {3 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 332, normalized size = 1.78 \begin {gather*} \frac {4 a c x+4 b \sqrt {1-c^2 x^2}+\frac {b \sqrt {1-c^2 x^2}}{-1+c x}-\frac {b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a c x}{-1+c^2 x^2}+3 i b \pi \text {ArcSin}(c x)+4 b c x \text {ArcSin}(c x)+\frac {b \text {ArcSin}(c x)}{1-c x}-\frac {b \text {ArcSin}(c x)}{1+c x}-3 b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-6 b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-3 b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+6 b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+3 a \log (1-c x)-3 a \log (1+c x)+3 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+3 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-6 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+6 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 273, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {\frac {a c x}{d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 d^{2}}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {b \arcsin \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {3 i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}}{c^{5}}\) | \(273\) |
default | \(\frac {\frac {a c x}{d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 d^{2}}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {b \arcsin \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {3 i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}}{c^{5}}\) | \(273\) |
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 x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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