Optimal. Leaf size=56 \[ \frac {3 \cos (c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{2 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {3 x}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2679, 2682, 8} \[ \frac {3 \cos (c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{2 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {3 x}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2679
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\cos ^3(c+d x)}{2 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {3 \int \frac {\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{2 a}\\ &=\frac {3 \cos (c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{2 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {3 \int 1 \, dx}{2 a^2}\\ &=\frac {3 x}{2 a^2}+\frac {3 \cos (c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{2 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 109, normalized size = 1.95 \[ -\frac {\left (\sqrt {\sin (c+d x)+1} \left (\sin ^2(c+d x)-5 \sin (c+d x)+4\right )-6 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}\right ) \cos ^5(c+d x)}{2 a^2 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 35, normalized size = 0.62 \[ \frac {3 \, d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 73, normalized size = 1.30 \[ \frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.19, size = 142, normalized size = 2.54 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.49, size = 140, normalized size = 2.50 \[ -\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 4}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.65, size = 32, normalized size = 0.57 \[ \frac {4\,\cos \left (c+d\,x\right )-\frac {\sin \left (2\,c+2\,d\,x\right )}{2}+3\,d\,x}{2\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 22.41, size = 403, normalized size = 7.20 \[ \begin {cases} \frac {3 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} + \frac {6 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} + \frac {3 d x}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} + \frac {2 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} + \frac {8 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} - \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} + \frac {8}{2 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________