Optimal. Leaf size=63 \[ -\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )-\frac {1}{2} i b \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4626, 3719, 2190, 2279, 2391} \[ -\frac {1}{2} i b \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4626
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x} \, dx &=-\operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-b \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )+\frac {1}{2} (i b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cos ^{-1}(c x)\right ) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {1}{2} i b \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 0.92 \[ a \log (x)-\frac {1}{2} i b \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )-\frac {1}{2} i b \cos ^{-1}(c x)^2+b \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arccos \left (c x\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \arccos \left (c x\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.22 \[ a \ln \left (c x \right )-\frac {i b \arccos \left (c x \right )^{2}}{2}+b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2} \]
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 \[ b \int \frac {\arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )}{x}\,{d x} + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {acos}{\left (c x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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