Optimal. Leaf size=172 \[ -\frac {4 b \sqrt {1-c^2 x^2}}{3 c^5 d}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {x (a+b \text {ArcSin}(c x))}{c^4 d}-\frac {x^3 (a+b \text {ArcSin}(c x))}{3 c^2 d}-\frac {2 i (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^5 d}+\frac {i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^5 d}-\frac {i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^5 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4795, 4749,
4266, 2317, 2438, 267, 272, 45} \begin {gather*} -\frac {2 i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^5 d}-\frac {x (a+b \text {ArcSin}(c x))}{c^4 d}-\frac {x^3 (a+b \text {ArcSin}(c x))}{3 c^2 d}+\frac {i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^5 d}-\frac {i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{c^5 d}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {4 b \sqrt {1-c^2 x^2}}{3 c^5 d} \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 4795
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac {\int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{3 c d}\\ &=-\frac {x \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{c^4}+\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^3 d}+\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{6 c d}\\ &=-\frac {b \sqrt {1-c^2 x^2}}{c^5 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac {\text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^5 d}+\frac {b \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{6 c d}\\ &=-\frac {4 b \sqrt {1-c^2 x^2}}{3 c^5 d}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d}-\frac {b \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^5 d}+\frac {b \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^5 d}\\ &=-\frac {4 b \sqrt {1-c^2 x^2}}{3 c^5 d}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^5 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^5 d}\\ &=-\frac {4 b \sqrt {1-c^2 x^2}}{3 c^5 d}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {x \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d}+\frac {i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^5 d}-\frac {i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 286, normalized size = 1.66 \begin {gather*} -\frac {18 a c x+6 a c^3 x^3+22 b \sqrt {1-c^2 x^2}+2 b c^2 x^2 \sqrt {1-c^2 x^2}+9 i b \pi \text {ArcSin}(c x)+18 b c x \text {ArcSin}(c x)+6 b c^3 x^3 \text {ArcSin}(c x)-9 b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-18 b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-9 b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+18 b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+9 a \log (1-c x)-9 a \log (1+c x)+9 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+9 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-18 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )+18 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{18 c^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 249, normalized size = 1.45
method | result | size |
derivativedivides | \(\frac {-\frac {a \,c^{3} x^{3}}{3 d}-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {11 b \sqrt {-c^{2} x^{2}+1}}{9 d}-\frac {b \arcsin \left (c x \right ) c x}{d}-\frac {b \arcsin \left (c x \right ) c^{3} x^{3}}{3 d}-\frac {b \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9 d}}{c^{5}}\) | \(249\) |
default | \(\frac {-\frac {a \,c^{3} x^{3}}{3 d}-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {11 b \sqrt {-c^{2} x^{2}+1}}{9 d}-\frac {b \arcsin \left (c x \right ) c x}{d}-\frac {b \arcsin \left (c x \right ) c^{3} x^{3}}{3 d}-\frac {b \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9 d}}{c^{5}}\) | \(249\) |
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^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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