Optimal. Leaf size=46 \[ \frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0520895, antiderivative size = 46, 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} \[ \frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{(a+x)^4}+\frac{1}{(a+x)^3}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{1}{3 a d (a+a \sin (c+d x))^3}-\frac{1}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.028238, size = 30, normalized size = 0.65 \[ -\frac{3 \sin (c+d x)+1}{6 a^4 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 33, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{4}} \left ( -{\frac{1}{2\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08795, size = 77, normalized size = 1.67 \begin{align*} -\frac{3 \, \sin \left (d x + c\right ) + 1}{6 \,{\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41514, size = 147, normalized size = 3.2 \begin{align*} \frac{3 \, \sin \left (d x + c\right ) + 1}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.86334, size = 129, normalized size = 2.8 \begin{align*} \begin{cases} - \frac{3 \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} - \frac{1}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1799, size = 38, normalized size = 0.83 \begin{align*} -\frac{3 \, \sin \left (d x + c\right ) + 1}{6 \, a^{4} d{\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]