Optimal. Leaf size=32 \[ \frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2667} \[ \frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}(\int (a-x) \, dx,x,a \sin (c+d x))}{a^3 d}\\ &=\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 24, normalized size = 0.75 \[ -\frac {(\sin (c+d x)-2) \sin (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 25, normalized size = 0.78 \[ \frac {\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 25, normalized size = 0.78 \[ -\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 28, normalized size = 0.88 \[ -\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 25, normalized size = 0.78 \[ -\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 22, normalized size = 0.69 \[ -\frac {\sin \left (c+d\,x\right )\,\left (\sin \left (c+d\,x\right )-2\right )}{2\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.87, size = 158, normalized size = 4.94 \[ \begin {cases} \frac {2 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {2 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________