Optimal. Leaf size=12 \[ \frac {\log (a-b \cos (x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2747, 31}
\begin {gather*} \frac {\log (a-b \cos (x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2747
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a-b \cos (x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,-b \cos (x)\right )}{b}\\ &=\frac {\log (a-b \cos (x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} \frac {\log (a-b \cos (x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 13, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\ln \left (a -b \cos \left (x \right )\right )}{b}\) | \(13\) |
default | \(\frac {\ln \left (a -b \cos \left (x \right )\right )}{b}\) | \(13\) |
risch | \(-\frac {i x}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\) | \(32\) |
norman | \(\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a -b \right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.90, size = 13, normalized size = 1.08 \begin {gather*} \frac {\log \left (b \cos \left (x\right ) - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.77, size = 12, normalized size = 1.00 \begin {gather*} \frac {\log \left (-b \cos \left (x\right ) + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.16, size = 15, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {\log {\left (- \frac {a}{b} + \cos {\left (x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\left (x \right )}}{a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.02, size = 14, normalized size = 1.17 \begin {gather*} \frac {\log \left ({\left | b \cos \left (x\right ) - a \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.20, size = 13, normalized size = 1.08 \begin {gather*} \frac {\ln \left (b\,\cos \left (x\right )-a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________