Optimal. Leaf size=32 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2707, 36, 29, 31} \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,a \sin (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\log (\sin (c+d x))}{a d}-\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 28, normalized size = 0.88 \[ \frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - \log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 33, normalized size = 1.03 \[ -\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 33, normalized size = 1.03 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 31, normalized size = 0.97 \[ -\frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.53, size = 32, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________