Optimal. Leaf size=235 \[ -\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \text {ArcSin}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))-\frac {i d^3 (a+b \text {ArcSin}(c x))^2}{2 b}+d^3 (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} i b d^3 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4773, 4721,
3798, 2221, 2317, 2438, 201, 222} \begin {gather*} \frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))-\frac {i d^3 (a+b \text {ArcSin}(c x))^2}{2 b}+d^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )-\frac {19}{48} b d^3 \text {ArcSin}(c x)-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4773
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{6} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^2 \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{36} \left (5 b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx\\ &=-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx-\frac {1}{48} \left (5 b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{96} \left (5 b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{32} \left (3 b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-\left (2 i d^3\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (b d^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \left (i b d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 207, normalized size = 0.88 \begin {gather*} -\frac {1}{144} d^3 \left (216 a c^2 x^2-108 a c^4 x^4+24 a c^6 x^6+75 b c x \sqrt {1-c^2 x^2}-22 b c^3 x^3 \sqrt {1-c^2 x^2}+4 b c^5 x^5 \sqrt {1-c^2 x^2}+72 i b \text {ArcSin}(c x)^2-150 b \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+12 b \text {ArcSin}(c x) \left (18 c^2 x^2-9 c^4 x^4+2 c^6 x^6-12 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )-144 a \log (x)+72 i b \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 266, normalized size = 1.13
method | result | size |
derivativedivides | \(-\frac {d^{3} a \,c^{6} x^{6}}{6}+\frac {3 d^{3} a \,c^{4} x^{4}}{4}-\frac {3 d^{3} a \,c^{2} x^{2}}{2}+d^{3} a \ln \left (c x \right )-\frac {i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}-\frac {d^{3} b \sin \left (6 \arcsin \left (c x \right )\right )}{1152}+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}-\frac {d^{3} b \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {29 d^{3} b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}-\frac {29 d^{3} b \sin \left (2 \arcsin \left (c x \right )\right )}{128}\) | \(266\) |
default | \(-\frac {d^{3} a \,c^{6} x^{6}}{6}+\frac {3 d^{3} a \,c^{4} x^{4}}{4}-\frac {3 d^{3} a \,c^{2} x^{2}}{2}+d^{3} a \ln \left (c x \right )-\frac {i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}-\frac {d^{3} b \sin \left (6 \arcsin \left (c x \right )\right )}{1152}+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}-\frac {d^{3} b \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {29 d^{3} b \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{64}-\frac {29 d^{3} b \sin \left (2 \arcsin \left (c x \right )\right )}{128}\) | \(266\) |
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*} - d^{3} \left (\int \left (- \frac {a}{x}\right )\, dx + \int 3 a c^{2} x\, dx + \int \left (- 3 a c^{4} x^{3}\right )\, dx + \int a c^{6} x^{5}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 3 b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{5} \operatorname {asin}{\left (c x \right )}\, dx\right ) \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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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