Optimal. Leaf size=210 \[ -\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{d x}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d x^2}-\frac {2 c^2 (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d}-\frac {i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{d}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )}{2 d} \]
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Rubi [A]
time = 0.27, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4789, 4769,
4504, 4268, 2611, 2320, 6724, 4771, 29} \begin {gather*} \frac {i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d}-\frac {i b c^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{d x}-\frac {2 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d x^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2320
Rule 2611
Rule 4268
Rule 4504
Rule 4769
Rule 4771
Rule 4789
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+c^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c^2 \text {Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x} \, dx}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {b^2 c^2 \log (x)}{d}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.79, size = 353, normalized size = 1.68 \begin {gather*} -\frac {\frac {a^2}{x^2}-2 a^2 c^2 \log (x)+a^2 c^2 \log \left (1-c^2 x^2\right )+2 a b c^2 \left (\frac {\sqrt {1-c^2 x^2}}{c x}+\frac {\text {ArcSin}(c x)}{c^2 x^2}-2 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+2 \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+2 b^2 c^2 \left (\frac {i \pi ^3}{24}+\frac {\sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c x}+\frac {\text {ArcSin}(c x)^2}{2 c^2 x^2}-\frac {2}{3} i \text {ArcSin}(c x)^3-\text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x)^2 \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-\log (c x)-i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )-i \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )\right )}{2 d} \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. 740 vs. \(2 (250 ) = 500\).
time = 0.43, size = 741, normalized size = 3.53
method | result | size |
derivativedivides | \(c^{2} \left (\frac {i b^{2} \arcsin \left (c x \right )}{d}+\frac {i a b}{d}-\frac {a^{2}}{2 d \,c^{2} x^{2}}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d c x}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d}-\frac {2 b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}-\frac {a^{2} \ln \left (c x +1\right )}{2 d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {a^{2} \ln \left (c x \right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{d c x}-\frac {a b \arcsin \left (c x \right )}{d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}\right )\) | \(741\) |
default | \(c^{2} \left (\frac {i b^{2} \arcsin \left (c x \right )}{d}+\frac {i a b}{d}-\frac {a^{2}}{2 d \,c^{2} x^{2}}+\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d c x}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d}-\frac {2 b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a^{2} \ln \left (c x -1\right )}{2 d}-\frac {a^{2} \ln \left (c x +1\right )}{2 d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {a^{2} \ln \left (c x \right )}{d}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{d c x}-\frac {a b \arcsin \left (c x \right )}{d \,c^{2} x^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}\right )\) | \(741\) |
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^{2}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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