Optimal. Leaf size=248 \[ -\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 (a+b \text {ArcSin}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {ArcSin}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {ArcSin}(c x))}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^3}+\frac {3 i b c^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.25, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4789, 4793,
4769, 4504, 4268, 2317, 2438, 197, 198, 277} \begin {gather*} \frac {3 c^2 (a+b \text {ArcSin}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 (a+b \text {ArcSin}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {ArcSin}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {6 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^3}+\frac {3 i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 277
Rule 2317
Rule 2438
Rule 4268
Rule 4504
Rule 4769
Rule 4789
Rule 4793
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {\left (2 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac {\left (4 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (6 c^2\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {\left (3 i b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {\left (3 i b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ &=-\frac {b c}{2 d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {3 i b c^2 \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac {3 i b c^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 1.06, size = 256, normalized size = 1.03 \begin {gather*} -\frac {\frac {6 a}{x^2}-\frac {3 a c^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 a c^2}{-1+c^2 x^2}-36 a c^2 \log (x)+18 a c^2 \log \left (1-c^2 x^2\right )+b c^2 \left (\frac {c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac {14 c x}{\sqrt {1-c^2 x^2}}+\frac {6 \sqrt {1-c^2 x^2}}{c x}+\frac {6 \text {ArcSin}(c x)}{c^2 x^2}-\frac {3 \text {ArcSin}(c x)}{\left (-1+c^2 x^2\right )^2}+\frac {12 \text {ArcSin}(c x)}{-1+c^2 x^2}-36 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+36 \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-18 i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+18 i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )}{12 d^3} \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. 599 vs. \(2 (257 ) = 514\).
time = 0.29, size = 600, normalized size = 2.42
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a}{16 d^{3} \left (c x +1\right )^{2}}+\frac {9 a}{16 d^{3} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{2 d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {9 a}{16 d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{2 d^{3}}-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 a \ln \left (c x \right )}{d^{3}}+\frac {4 i b \,c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) c^{2} x^{2}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 i b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {b c x \sqrt {-c^{2} x^{2}+1}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {9 b \arcsin \left (c x \right )}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 i b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c x}-\frac {b \arcsin \left (c x \right )}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {2 i b}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{3}}+\frac {3 i b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{3}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {2 i b \,c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}\right )\) | \(600\) |
default | \(c^{2} \left (\frac {a}{16 d^{3} \left (c x +1\right )^{2}}+\frac {9 a}{16 d^{3} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{2 d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {9 a}{16 d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{2 d^{3}}-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 a \ln \left (c x \right )}{d^{3}}+\frac {4 i b \,c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) c^{2} x^{2}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 i b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {b c x \sqrt {-c^{2} x^{2}+1}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {9 b \arcsin \left (c x \right )}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 i b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c x}-\frac {b \arcsin \left (c x \right )}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {2 i b}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{3}}+\frac {3 i b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{3}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{3}}-\frac {2 i b \,c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}\right )\) | \(600\) |
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}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx}{d^{3}} \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 {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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