Optimal. Leaf size=63 \[ -\frac {i (a+b \text {ArcCos}(c x))^2}{2 b}+(a+b \text {ArcCos}(c x)) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )-\frac {1}{2} i b \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right ) \]
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Rubi [A]
time = 0.05, 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 = {4722, 3800,
2221, 2317, 2438} \begin {gather*} -\frac {i (a+b \text {ArcCos}(c x))^2}{2 b}+\log \left (1+e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-\frac {1}{2} i b \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4722
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x} \, dx &=-\text {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 \text {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 \text {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) \text {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 \begin {gather*} -\frac {1}{2} i b \text {ArcCos}(c x)^2+b \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+a \log (x)-\frac {1}{2} i b \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 77, normalized size = 1.22
method | result | size |
derivativedivides | \(a \ln \left (c x \right )-\frac {i b \arccos \left (c x \right )^{2}}{2}+b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(77\) |
default | \(a \ln \left (c x \right )-\frac {i b \arccos \left (c x \right )^{2}}{2}+b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\) | \(77\) |
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*} \int \frac {a + b \operatorname {acos}{\left (c x \right )}}{x}\, dx \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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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