Integrand size = 29, antiderivative size = 89 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {a \cos ^7(e+f x)}{18 f (a+a \sin (e+f x))^8}+\frac {25 \cos ^5(e+f x)}{126 a f (a+a \sin (e+f x))^6}-\frac {47 \cos ^5(e+f x)}{315 a^2 f (a+a \sin (e+f x))^5} \]
[Out]
Time = 0.35 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2954, 2951, 2729, 2727} \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {47 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)}+\frac {268 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)^2}-\frac {181 \cos (e+f x)}{105 a^7 f (\sin (e+f x)+1)^3}+\frac {92 \cos (e+f x)}{63 a^7 f (\sin (e+f x)+1)^4}-\frac {4 \cos (e+f x)}{9 a^7 f (\sin (e+f x)+1)^5} \]
[In]
[Out]
Rule 2727
Rule 2729
Rule 2951
Rule 2954
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^8(e+f x) (a-a \sin (e+f x))^7 \tan ^2(e+f x) \, dx}{a^{14}} \\ & = \frac {\int \left (\frac {4}{a^3 (1+\sin (e+f x))^5}-\frac {12}{a^3 (1+\sin (e+f x))^4}+\frac {13}{a^3 (1+\sin (e+f x))^3}-\frac {6}{a^3 (1+\sin (e+f x))^2}+\frac {1}{a^3 (1+\sin (e+f x))}\right ) \, dx}{a^4} \\ & = \frac {\int \frac {1}{1+\sin (e+f x)} \, dx}{a^7}+\frac {4 \int \frac {1}{(1+\sin (e+f x))^5} \, dx}{a^7}-\frac {6 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{a^7}-\frac {12 \int \frac {1}{(1+\sin (e+f x))^4} \, dx}{a^7}+\frac {13 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{a^7} \\ & = -\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {12 \cos (e+f x)}{7 a^7 f (1+\sin (e+f x))^4}-\frac {13 \cos (e+f x)}{5 a^7 f (1+\sin (e+f x))^3}+\frac {2 \cos (e+f x)}{a^7 f (1+\sin (e+f x))^2}-\frac {\cos (e+f x)}{a^7 f (1+\sin (e+f x))}+\frac {16 \int \frac {1}{(1+\sin (e+f x))^4} \, dx}{9 a^7}-\frac {2 \int \frac {1}{1+\sin (e+f x)} \, dx}{a^7}-\frac {36 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{7 a^7}+\frac {26 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{5 a^7} \\ & = -\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {11 \cos (e+f x)}{7 a^7 f (1+\sin (e+f x))^3}+\frac {4 \cos (e+f x)}{15 a^7 f (1+\sin (e+f x))^2}+\frac {\cos (e+f x)}{a^7 f (1+\sin (e+f x))}+\frac {16 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{21 a^7}+\frac {26 \int \frac {1}{1+\sin (e+f x)} \, dx}{15 a^7}-\frac {72 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{35 a^7} \\ & = -\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {20 \cos (e+f x)}{21 a^7 f (1+\sin (e+f x))^2}-\frac {11 \cos (e+f x)}{15 a^7 f (1+\sin (e+f x))}+\frac {32 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{105 a^7}-\frac {24 \int \frac {1}{1+\sin (e+f x)} \, dx}{35 a^7} \\ & = -\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {268 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))^2}-\frac {\cos (e+f x)}{21 a^7 f (1+\sin (e+f x))}+\frac {32 \int \frac {1}{1+\sin (e+f x)} \, dx}{315 a^7} \\ & = -\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {268 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))^2}-\frac {47 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(293\) vs. \(2(89)=178\).
Time = 3.16 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=\frac {1890 \cos \left (\frac {f x}{2}\right )+718830 \cos \left (e+\frac {f x}{2}\right )-467208 \cos \left (e+\frac {3 f x}{2}\right )-1260 \cos \left (2 e+\frac {3 f x}{2}\right )-540 \cos \left (2 e+\frac {5 f x}{2}\right )-179640 \cos \left (3 e+\frac {5 f x}{2}\right )+30753 \cos \left (3 e+\frac {7 f x}{2}\right )+135 \cos \left (4 e+\frac {7 f x}{2}\right )+15 \cos \left (4 e+\frac {9 f x}{2}\right )-15 \cos \left (5 e+\frac {9 f x}{2}\right )+971082 \sin \left (\frac {f x}{2}\right )+1890 \sin \left (e+\frac {f x}{2}\right )+1260 \sin \left (e+\frac {3 f x}{2}\right )+659400 \sin \left (2 e+\frac {3 f x}{2}\right )-303192 \sin \left (2 e+\frac {5 f x}{2}\right )-540 \sin \left (3 e+\frac {5 f x}{2}\right )-135 \sin \left (3 e+\frac {7 f x}{2}\right )-89955 \sin \left (4 e+\frac {7 f x}{2}\right )+13427 \sin \left (4 e+\frac {9 f x}{2}\right )+15 \sin \left (5 e+\frac {9 f x}{2}\right )}{720720 a^7 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
[In]
[Out]
Time = 0.63 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(\frac {-\frac {4}{315}-\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {16 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+4 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {8 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {28 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}}{f \,a^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}\) | \(100\) |
derivativedivides | \(\frac {-\frac {328}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {8}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}+\frac {352}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {832}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}}{f \,a^{7}}\) | \(115\) |
default | \(\frac {-\frac {328}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {8}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}+\frac {352}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {832}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}}{f \,a^{7}}\) | \(115\) |
risch | \(-\frac {2 \left (-2520 i {\mathrm e}^{5 i \left (f x +e \right )}-2310 \,{\mathrm e}^{6 i \left (f x +e \right )}+3402 \,{\mathrm e}^{4 i \left (f x +e \right )}+630 i {\mathrm e}^{7 i \left (f x +e \right )}+315 \,{\mathrm e}^{8 i \left (f x +e \right )}+1638 i {\mathrm e}^{3 i \left (f x +e \right )}-1062 \,{\mathrm e}^{2 i \left (f x +e \right )}-108 i {\mathrm e}^{i \left (f x +e \right )}+47\right )}{315 f \,a^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{9}}\) | \(117\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (83) = 166\).
Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.73 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} - 115 \, \cos \left (f x + e\right )^{3} - 265 \, \cos \left (f x + e\right )^{2} - {\left (47 \, \cos \left (f x + e\right )^{4} - 80 \, \cos \left (f x + e\right )^{3} - 195 \, \cos \left (f x + e\right )^{2} + 70 \, \cos \left (f x + e\right ) + 140\right )} \sin \left (f x + e\right ) + 70 \, \cos \left (f x + e\right ) + 140}{315 \, {\left (a^{7} f \cos \left (f x + e\right )^{5} + 5 \, a^{7} f \cos \left (f x + e\right )^{4} - 8 \, a^{7} f \cos \left (f x + e\right )^{3} - 20 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f + {\left (a^{7} f \cos \left (f x + e\right )^{4} - 4 \, a^{7} f \cos \left (f x + e\right )^{3} - 12 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f\right )} \sin \left (f x + e\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (83) = 166\).
Time = 0.23 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.76 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {126 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {315 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {210 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (a^{7} + \frac {9 \, a^{7} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{7} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{7} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \]
[In]
[Out]
none
Time = 0.62 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4 \, {\left (210 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 126 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{315 \, a^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \]
[In]
[Out]
Time = 10.49 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+9\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-126\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+210\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{315\,a^7\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \]
[In]
[Out]