Optimal. Leaf size=83 \[ \frac {\sin (c+d x)}{5 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {\sin (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2750, 2650, 2648} \[ \frac {\sin (c+d x)}{5 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {\sin (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rule 2650
Rule 2750
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {3 \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{5 a}\\ &=-\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {1}{a+a \cos (c+d x)} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {\sin (c+d x)}{5 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 71, normalized size = 0.86 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-5 \sin \left (c+\frac {d x}{2}\right )+5 \sin \left (c+\frac {3 d x}{2}\right )+\sin \left (2 c+\frac {5 d x}{2}\right )+5 \sin \left (\frac {d x}{2}\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{80 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 73, normalized size = 0.88 \[ \frac {{\left (\cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 31, normalized size = 0.37 \[ -\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{20 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 32, normalized size = 0.39 \[ \frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 47, normalized size = 0.57 \[ \frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{20 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 30, normalized size = 0.36 \[ -\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-5\right )}{20\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.28, size = 48, normalized size = 0.58 \[ \begin {cases} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________