Optimal. Leaf size=34 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (a+b \sin (c+d x))}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 34, 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 = {2721, 36, 29, 31} \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (a+b \sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2721
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,b \sin (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (c+d x)\right )}{a d}\\ &=\frac {\log (\sin (c+d x))}{a d}-\frac {\log (a+b \sin (c+d x))}{a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 34, normalized size = 1.00 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (a+b \sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 31, normalized size = 0.91 \[ -\frac {\log \left (b \sin \left (d x + c\right ) + a\right ) - \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 35, normalized size = 1.03 \[ -\frac {\frac {\log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.22, size = 35, normalized size = 1.03 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 33, normalized size = 0.97 \[ -\frac {\frac {\log \left (b \sin \left (d x + c\right ) + a\right )}{a} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 11.83, size = 48, normalized size = 1.41 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )} \csc {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________