Optimal. Leaf size=76 \[ \frac {2 (a+b \text {ArcCos}(c x)) \tanh ^{-1}\left (e^{i \text {ArcCos}(c x)}\right )}{c d}-\frac {i b \text {PolyLog}\left (2,-e^{i \text {ArcCos}(c x)}\right )}{c d}+\frac {i b \text {PolyLog}\left (2,e^{i \text {ArcCos}(c x)}\right )}{c d} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4750, 4268,
2317, 2438} \begin {gather*} \frac {2 \tanh ^{-1}\left (e^{i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))}{c d}-\frac {i b \text {Li}_2\left (-e^{i \text {ArcCos}(c x)}\right )}{c d}+\frac {i b \text {Li}_2\left (e^{i \text {ArcCos}(c x)}\right )}{c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4268
Rule 4750
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx &=-\frac {\text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{c d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c d}-\frac {b \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c d}\\ &=\frac {2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d}-\frac {i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac {i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{c d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 107, normalized size = 1.41 \begin {gather*} \frac {-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 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 156, normalized size = 2.05
method | result | size |
derivativedivides | \(\frac {\frac {a \arctanh \left (c x \right )}{d}+\frac {i b \arctanh \left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}-\frac {i b \arctanh \left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}+\frac {b \arctanh \left (c x \right ) \arccos \left (c x \right )}{d}-\frac {i b \dilog \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}+\frac {i b \dilog \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}}{c}\) | \(156\) |
default | \(\frac {\frac {a \arctanh \left (c x \right )}{d}+\frac {i b \arctanh \left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}-\frac {i b \arctanh \left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}+\frac {b \arctanh \left (c x \right ) \arccos \left (c x \right )}{d}-\frac {i b \dilog \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}+\frac {i b \dilog \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{d}}{c}\) | \(156\) |
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^{2} - 1}\, dx + \int \frac {b \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 {a+b\,\mathrm {acos}\left (c\,x\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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