Optimal. Leaf size=58 \[ \frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2859, 2671} \[ \frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2671
Rule 2859
Rubi steps
\begin {align*} \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx &=\frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}+\frac {6 \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^5} \, dx}{7 a}\\ &=\frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a+a \sin (e+f x))^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.26, size = 143, normalized size = 2.47 \[ \frac {1134 \sin \left (2 e+\frac {3 f x}{2}\right )-224 \sin \left (2 e+\frac {5 f x}{2}\right )+\sin \left (4 e+\frac {7 f x}{2}\right )+4585 \cos \left (e+\frac {f x}{2}\right )-2982 \cos \left (e+\frac {3 f x}{2}\right )-1148 \cos \left (3 e+\frac {5 f x}{2}\right )+197 \cos \left (3 e+\frac {7 f x}{2}\right )+2275 \sin \left (\frac {f x}{2}\right )}{4620 a^6 f \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 195, normalized size = 3.36 \[ \frac {6 \, \cos \left (f x + e\right )^{4} - 11 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )^{2} + {\left (6 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} - 10 \, \cos \left (f x + e\right ) - 20\right )} \sin \left (f x + e\right ) + 10 \, \cos \left (f x + e\right ) + 20}{35 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} - 3 \, a^{6} f \cos \left (f x + e\right )^{3} - 8 \, a^{6} f \cos \left (f x + e\right )^{2} + 4 \, a^{6} f \cos \left (f x + e\right ) + 8 \, a^{6} f - {\left (a^{6} f \cos \left (f x + e\right )^{3} + 4 \, a^{6} f \cos \left (f x + e\right )^{2} - 4 \, a^{6} f \cos \left (f x + e\right ) - 8 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 92, normalized size = 1.59 \[ -\frac {2 \, {\left (35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 14 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{35 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 100, normalized size = 1.72 \[ \frac {\frac {224}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.52, size = 269, normalized size = 4.64 \[ -\frac {2 \, {\left (\frac {7 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {14 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + 1\right )}}{35 \, {\left (a^{6} + \frac {7 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.89, size = 157, normalized size = 2.71 \[ -\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-14\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+70\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-35\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+35\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}{35\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________