Optimal. Leaf size=71 \[ \frac {2 (a+b \text {ArcCos}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcCos}(c x)}\right )}{d}-\frac {i b \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )}{2 d}+\frac {i b \text {PolyLog}\left (2,e^{2 i \text {ArcCos}(c x)}\right )}{2 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4770, 4504,
4268, 2317, 2438} \begin {gather*} \frac {2 \tanh ^{-1}\left (e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))}{d}-\frac {i b \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right )}{2 d}+\frac {i b \text {Li}_2\left (e^{2 i \text {ArcCos}(c x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4268
Rule 4504
Rule 4770
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx &=-\frac {\text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=-\frac {2 \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}-\frac {i b \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {i b \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 104, normalized size = 1.46 \begin {gather*} -\frac {2 b \text {ArcCos}(c x) \log \left (1-e^{2 i \text {ArcCos}(c x)}\right )-2 b \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )-2 a \log (x)+a \log \left (1-c^2 x^2\right )+i b \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )-i b \text {PolyLog}\left (2,e^{2 i \text {ArcCos}(c x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 168, normalized size = 2.37
method | result | size |
derivativedivides | \(-\frac {a \ln \left (c x +1\right )}{2 d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x \right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {i b \dilog \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b \dilog \left (1-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}\) | \(168\) |
default | \(-\frac {a \ln \left (c x +1\right )}{2 d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x \right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {i b \dilog \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {i b \dilog \left (1-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}\) | \(168\) |
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^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{3} - x}\, 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 {a+b\,\mathrm {acos}\left (c\,x\right )}{x\,\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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