Optimal. Leaf size=31 \[ \frac {\sin (c+d x)}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {\sin (c+d x)}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 25, normalized size = 0.81 \[ \frac {\sin (c+d x)-\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 26, normalized size = 0.84 \[ -\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 31, normalized size = 1.00 \[ -\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 32, normalized size = 1.03 \[ -\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a d}+\frac {\sin \left (d x +c \right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 30, normalized size = 0.97 \[ -\frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.46, size = 26, normalized size = 0.84 \[ -\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )-\sin \left (c+d\,x\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.68, size = 37, normalized size = 1.19 \[ \begin {cases} - \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________