Optimal. Leaf size=11 \[ \frac {\log (\sin (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3556}
\begin {gather*} \frac {\log (\sin (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rubi steps
\begin {align*} \int \cot (a+b x) \, dx &=\frac {\log (\sin (a+b x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 19, normalized size = 1.73 \begin {gather*} \frac {\log (\cos (a+b x))+\log (\tan (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(25\) vs.
\(2(11)=22\).
time = 0.01, size = 26, normalized size = 2.36
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (1+\tan ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(26\) |
default | \(\frac {-\frac {\ln \left (1+\tan ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(26\) |
norman | \(\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (b x +a \right )\right )}{2 b}\) | \(29\) |
risch | \(-i x -\frac {2 i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.54, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (11) = 22\).
time = 0.75, size = 27, normalized size = 2.45 \begin {gather*} \frac {\log \left (\frac {\tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (8) = 16\).
time = 0.16, size = 29, normalized size = 2.64 \begin {gather*} \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} + \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\tan {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (11) = 22\).
time = 1.19, size = 56, normalized size = 5.09 \begin {gather*} \frac {\log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.19, size = 28, normalized size = 2.55 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________