Optimal. Leaf size=59 \[ -\frac {1}{2} i \text {ArcCos}\left (\frac {x}{a}\right )^2+\text {ArcCos}\left (\frac {x}{a}\right ) \log \left (1+e^{2 i \text {ArcCos}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}\left (\frac {x}{a}\right )}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5372, 4722,
3800, 2221, 2317, 2438} \begin {gather*} -\frac {1}{2} i \text {Li}_2\left (-e^{2 i \text {ArcCos}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \text {ArcCos}\left (\frac {x}{a}\right )^2+\text {ArcCos}\left (\frac {x}{a}\right ) \log \left (1+e^{2 i \text {ArcCos}\left (\frac {x}{a}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4722
Rule 5372
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x} \, dx &=\int \frac {\cos ^{-1}\left (\frac {x}{a}\right )}{x} \, dx\\ &=-\text {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \cos ^{-1}\left (\frac {x}{a}\right )^2+2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \cos ^{-1}\left (\frac {x}{a}\right )^2+\cos ^{-1}\left (\frac {x}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {x}{a}\right )}\right )-\text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac {x}{a}\right )\right )\\ &=-\frac {1}{2} i \cos ^{-1}\left (\frac {x}{a}\right )^2+\cos ^{-1}\left (\frac {x}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {x}{a}\right )}\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac {x}{a}\right )}\right )\\ &=-\frac {1}{2} i \cos ^{-1}\left (\frac {x}{a}\right )^2+\cos ^{-1}\left (\frac {x}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac {x}{a}\right )}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 59, normalized size = 1.00 \begin {gather*} -\frac {1}{2} i \sec ^{-1}\left (\frac {a}{x}\right )^2+\sec ^{-1}\left (\frac {a}{x}\right ) \log \left (1+e^{2 i \sec ^{-1}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (\frac {a}{x}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.93, size = 76, normalized size = 1.29
method | result | size |
derivativedivides | \(-\frac {i \mathrm {arcsec}\left (\frac {a}{x}\right )^{2}}{2}+\mathrm {arcsec}\left (\frac {a}{x}\right ) \ln \left (1+\left (\frac {x}{a}+i \sqrt {1-\frac {x^{2}}{a^{2}}}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (\frac {x}{a}+i \sqrt {1-\frac {x^{2}}{a^{2}}}\right )^{2}\right )}{2}\) | \(76\) |
default | \(-\frac {i \mathrm {arcsec}\left (\frac {a}{x}\right )^{2}}{2}+\mathrm {arcsec}\left (\frac {a}{x}\right ) \ln \left (1+\left (\frac {x}{a}+i \sqrt {1-\frac {x^{2}}{a^{2}}}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (\frac {x}{a}+i \sqrt {1-\frac {x^{2}}{a^{2}}}\right )^{2}\right )}{2}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________