Optimal. Leaf size=49 \[ \frac{a (A+B) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{a B \sin ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0631775, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2833, 43} \[ \frac{a (A+B) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{a B \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a+x) \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a A+(A+B) x+\frac{B x^2}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a A \sin (c+d x)}{d}+\frac{a (A+B) \sin ^2(c+d x)}{2 d}+\frac{a B \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.394483, size = 46, normalized size = 0.94 \[ -\frac{a (\cos (2 (c+d x)) (3 (A+B)+2 B \sin (c+d x))-2 (6 A+B) \sin (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{aB \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( aA+aB \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}+A\sin \left ( dx+c \right ) a \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00871, size = 57, normalized size = 1.16 \begin{align*} \frac{2 \, B a \sin \left (d x + c\right )^{3} + 3 \,{\left (A + B\right )} a \sin \left (d x + c\right )^{2} + 6 \, A a \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64018, size = 120, normalized size = 2.45 \begin{align*} -\frac{3 \,{\left (A + B\right )} a \cos \left (d x + c\right )^{2} + 2 \,{\left (B a \cos \left (d x + c\right )^{2} -{\left (3 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.630478, size = 75, normalized size = 1.53 \begin{align*} \begin{cases} \frac{A a \sin{\left (c + d x \right )}}{d} - \frac{A a \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{B a \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{B a \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a \sin{\left (c \right )} + a\right ) \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26491, size = 70, normalized size = 1.43 \begin{align*} \frac{2 \, B a \sin \left (d x + c\right )^{3} + 3 \, A a \sin \left (d x + c\right )^{2} + 3 \, B a \sin \left (d x + c\right )^{2} + 6 \, A a \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]