Optimal. Leaf size=45 \[ \frac{(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0453911, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)^3}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a (a+x)^3+(a+x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{(a+a \sin (c+d x))^4}{4 a d}+\frac{(a+a \sin (c+d x))^5}{5 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.118371, size = 30, normalized size = 0.67 \[ \frac{a^3 (\sin (c+d x)+1)^4 (4 \sin (c+d x)-1)}{20 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 57, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15551, size = 78, normalized size = 1.73 \begin{align*} \frac{4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69621, size = 169, normalized size = 3.76 \begin{align*} \frac{15 \, a^{3} \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 7 \, a^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.39767, size = 102, normalized size = 2.27 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \sin{\left (c \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 = 1.27019, size = 78, normalized size = 1.73 \begin{align*} \frac{4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]