Optimal. Leaf size=115 \[ \frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \text {ArcCos}(c x))}{c^2 d}+\frac {2 (a+b \text {ArcCos}(c x)) \tanh ^{-1}\left (e^{i \text {ArcCos}(c x)}\right )}{c^3 d}-\frac {i b \text {PolyLog}\left (2,-e^{i \text {ArcCos}(c x)}\right )}{c^3 d}+\frac {i b \text {PolyLog}\left (2,e^{i \text {ArcCos}(c x)}\right )}{c^3 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4796, 4750,
4268, 2317, 2438, 267} \begin {gather*} \frac {2 \tanh ^{-1}\left (e^{i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))}{c^3 d}-\frac {x (a+b \text {ArcCos}(c x))}{c^2 d}-\frac {i b \text {Li}_2\left (-e^{i \text {ArcCos}(c x)}\right )}{c^3 d}+\frac {i b \text {Li}_2\left (e^{i \text {ArcCos}(c x)}\right )}{c^3 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2438
Rule 4268
Rule 4750
Rule 4796
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac {x \left (a+b \cos ^{-1}(c x)\right )}{c^2 d}+\frac {\int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c d}\\ &=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x \left (a+b \cos ^{-1}(c x)\right )}{c^2 d}-\frac {\text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x \left (a+b \cos ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^3 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^3 d}-\frac {b \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x \left (a+b \cos ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c^3 d}\\ &=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x \left (a+b \cos ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^3 d}-\frac {i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{c^3 d}+\frac {i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{c^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 138, normalized size = 1.20 \begin {gather*} -\frac {2 a c x-2 b \sqrt {1-c^2 x^2}+2 b c x \text {ArcCos}(c x)+2 b \text {ArcCos}(c x) \log \left (1-e^{i \text {ArcCos}(c x)}\right )-2 b \text {ArcCos}(c x) \log \left (1+e^{i \text {ArcCos}(c x)}\right )+a \log (1-c x)-a \log (1+c x)+2 i b \text {PolyLog}\left (2,-e^{i \text {ArcCos}(c x)}\right )-2 i b \text {PolyLog}\left (2,e^{i \text {ArcCos}(c x)}\right )}{2 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 186, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}+\frac {i b \dilog \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i b \dilog \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}}{c^{3}}\) | \(186\) |
default | \(\frac {-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}+\frac {i b \dilog \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i b \dilog \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}}{c^{3}}\) | \(186\) |
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 x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{2} \operatorname {acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, 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 = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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