Optimal. Leaf size=263 \[ \frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \text {ArcSin}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {3 i c^2 d^3 (a+b \text {ArcSin}(c x))^2}{2 b}-3 c^2 d^3 (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+\frac {3}{2} i b c^2 d^3 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4775, 283,
201, 222, 4773, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{2 x^2}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))+\frac {3 i c^2 d^3 (a+b \text {ArcSin}(c x))^2}{2 b}-3 c^2 d^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+\frac {3}{2} i b c^2 d^3 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )+\frac {3}{32} b c^2 d^3 \text {ArcSin}(c x)-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}+\frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4773
Rule 4775
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac {1}{2} \left (b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2} \, dx\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx-\frac {1}{2} \left (5 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx\\ &=-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx+\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \, dx+\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \, dx-\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \, dx\\ &=\frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (6 i c^2 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 {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\left (3 b c^2 d^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 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 (3 i b c^2 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac {3}{32} b c^3 d^3 x \sqrt {1-c^2 x^2}-\frac {7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} i b c^2 d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 226, normalized size = 0.86 \begin {gather*} -\frac {d^3 \left (16 a-48 a c^4 x^4+8 a c^6 x^6+16 b c x \sqrt {1-c^2 x^2}-21 b c^3 x^3 \sqrt {1-c^2 x^2}+2 b c^5 x^5 \sqrt {1-c^2 x^2}-48 i b c^2 x^2 \text {ArcSin}(c x)^2+42 b c^2 x^2 \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+8 b \text {ArcSin}(c x) \left (2-6 c^4 x^4+c^6 x^6+12 c^2 x^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )+96 a c^2 x^2 \log (x)-48 i b c^2 x^2 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )}{32 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 306, normalized size = 1.16
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {d^{3} a \,c^{4} x^{4}}{4}+\frac {3 d^{3} a \,c^{2} x^{2}}{2}-\frac {d^{3} a}{2 c^{2} x^{2}}-3 d^{3} a \ln \left (c x \right )+\frac {3 i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+\frac {5 b c \,d^{3} x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {5 d^{3} b \arcsin \left (c x \right ) c^{2} x^{2}}{4}-\frac {5 b \,d^{3} \arcsin \left (c x \right )}{8}+3 i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {d^{3} b \arcsin \left (c x \right )}{2 c^{2} x^{2}}-3 d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {i d^{3} b}{2}-\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}+\frac {d^{3} b \sin \left (4 \arcsin \left (c x \right )\right )}{128}\right )\) | \(306\) |
default | \(c^{2} \left (-\frac {d^{3} a \,c^{4} x^{4}}{4}+\frac {3 d^{3} a \,c^{2} x^{2}}{2}-\frac {d^{3} a}{2 c^{2} x^{2}}-3 d^{3} a \ln \left (c x \right )+\frac {3 i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+\frac {5 b c \,d^{3} x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {5 d^{3} b \arcsin \left (c x \right ) c^{2} x^{2}}{4}-\frac {5 b \,d^{3} \arcsin \left (c x \right )}{8}+3 i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {d^{3} b \arcsin \left (c x \right )}{2 c^{2} x^{2}}-3 d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {i d^{3} b}{2}-\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}+\frac {d^{3} b \sin \left (4 \arcsin \left (c x \right )\right )}{128}\right )\) | \(306\) |
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^{3}}\right )\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int \left (- 3 a c^{4} x\right )\, dx + \int a c^{6} x^{3}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b c^{4} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{3} \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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