Optimal. Leaf size=80 \[ \frac {9 \cos (c+d x)}{2 a^3 d}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {9 x}{2 a^3}+\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2859, 2679, 2682, 8} \[ \frac {9 \cos (c+d x)}{2 a^3 d}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {9 x}{2 a^3}+\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2679
Rule 2682
Rule 2859
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a}\\ &=\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {9 \int \frac {\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{2 a^2}\\ &=\frac {9 \cos (c+d x)}{2 a^3 d}+\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {9 \int 1 \, dx}{2 a^3}\\ &=\frac {9 x}{2 a^3}+\frac {9 \cos (c+d x)}{2 a^3 d}+\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.64, size = 143, normalized size = 1.79 \[ \frac {180 d x \sin \left (c+\frac {d x}{2}\right )+55 \sin \left (2 c+\frac {3 d x}{2}\right )-5 \sin \left (2 c+\frac {5 d x}{2}\right )+59 \cos \left (c+\frac {d x}{2}\right )+55 \cos \left (c+\frac {3 d x}{2}\right )+5 \cos \left (3 c+\frac {5 d x}{2}\right )-381 \sin \left (\frac {d x}{2}\right )+180 d x \cos \left (\frac {d x}{2}\right )}{40 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 100, normalized size = 1.25 \[ \frac {\cos \left (d x + c\right )^{3} + 9 \, d x + {\left (9 \, d x + 13\right )} \cos \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} + {\left (9 \, d x - \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) + 8}{2 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 91, normalized size = 1.14 \[ \frac {\frac {9 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {16}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.40, size = 163, normalized size = 2.04 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}+\frac {8}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 225, normalized size = 2.81 \[ \frac {\frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 14}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.80, size = 94, normalized size = 1.18 \[ \frac {9\,x}{2\,a^3}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 61.27, size = 1244, normalized size = 15.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________