Optimal. Leaf size=141 \[ -\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \text {ArcSin}(c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c d^2}+\frac {i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{2 c d^2}-\frac {i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{2 c d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4747, 4749,
4266, 2317, 2438, 267} \begin {gather*} -\frac {i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c d^2}+\frac {x (a+b \text {ArcSin}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{2 c d^2}-\frac {i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{2 c d^2}-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2438
Rule 4266
Rule 4747
Rule 4749
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}-\frac {b \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}+\frac {b \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c d^2}+\frac {i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}-\frac {i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 c d^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(334\) vs. \(2(141)=282\).
time = 0.40, size = 334, normalized size = 2.37 \begin {gather*} -\frac {\frac {b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {b \sqrt {1-c^2 x^2}}{c+c^2 x}+\frac {2 a x}{-1+c^2 x^2}+\frac {i b \pi \text {ArcSin}(c x)}{c}+\frac {b \text {ArcSin}(c x)}{c (-1+c x)}+\frac {b \text {ArcSin}(c x)}{c+c^2 x}-\frac {b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {2 b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {2 b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {a \log (1-c x)}{c}-\frac {a \log (1+c x)}{c}+\frac {b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}+\frac {b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}-\frac {2 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {2 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c}}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 238, normalized size = 1.69
method | result | size |
derivativedivides | \(\frac {-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 d^{2}}-\frac {b \arcsin \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}}{c}\) | \(238\) |
default | \(\frac {-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 d^{2}}-\frac {b \arcsin \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}}{c}\) | \(238\) |
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^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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