Integrand size = 25, antiderivative size = 215 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {27}{8} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) x+\frac {9 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac {9 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \cos (e+f x) \sin (e+f x)}{40 f}+\frac {9 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {9 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {9 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2842, 2832, 2813} \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {3}{8} a^2 x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )+\frac {a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f} \]
[In]
[Out]
Rule 2813
Rule 2832
Rule 2842
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int \left (9 a^2 d-a^2 (c-10 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{5 d} \\ & = \frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^2 d (11 c+10 d)-3 a^2 \left (c^2-10 c d-12 d^2\right ) \sin (e+f x)\right ) \, dx}{20 d} \\ & = \frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 a^2 d \left (31 c^2+50 c d+24 d^2\right )-3 a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x)\right ) \, dx}{60 d} \\ & = \frac {3}{8} a^2 (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) x+\frac {a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac {a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \cos (e+f x) \sin (e+f x)}{40 f}+\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {9 \cos (e+f x) \left (-8 \left (10 c^3+25 c^2 d+20 c d^2+6 d^3\right )-\frac {30 \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-5 \left (4 c^3+24 c^2 d+21 c d^2+6 d^3\right ) \sin (e+f x)-8 d \left (5 c^2+10 c d+3 d^2\right ) \sin ^2(e+f x)-10 d^2 (3 c+2 d) \sin ^3(e+f x)-8 d^3 \sin ^4(e+f x)\right )}{40 f} \]
[In]
[Out]
Time = 2.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (\left (c^{3}+6 c^{2} d +6 c \,d^{2}+2 d^{3}\right ) \sin \left (2 f x +2 e \right )-\left (c +\frac {3 d}{2}\right ) d \left (c +\frac {d}{2}\right ) \cos \left (3 f x +3 e \right )-\frac {3 d^{2} \left (c +\frac {2 d}{3}\right ) \sin \left (4 f x +4 e \right )}{8}+\frac {d^{3} \cos \left (5 f x +5 e \right )}{20}+\left (8 c^{3}+21 c^{2} d +18 c \,d^{2}+\frac {11}{2} d^{3}\right ) \cos \left (f x +e \right )+\left (-3 f x +\frac {24}{5}\right ) d^{3}+c \left (-\frac {21 f x}{2}+16\right ) d^{2}+\left (-12 f x +20\right ) c^{2} d +\left (-6 f x +8\right ) c^{3}\right )}{4 f}\) | \(168\) |
parts | \(a^{2} c^{3} x +\frac {\left (3 a^{2} c \,d^{2}+2 a^{2} d^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 a^{2} c^{3}+3 a^{2} c^{2} d \right ) \cos \left (f x +e \right )}{f}-\frac {\left (3 a^{2} c^{2} d +6 a^{2} c \,d^{2}+a^{2} d^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (a^{2} c^{3}+6 a^{2} c^{2} d +3 a^{2} c \,d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(230\) |
risch | \(\frac {3 a^{2} c^{3} x}{2}+3 a^{2} c^{2} d x +\frac {21 a^{2} c \,d^{2} x}{8}+\frac {3 a^{2} d^{3} x}{4}-\frac {2 a^{2} \cos \left (f x +e \right ) c^{3}}{f}-\frac {21 a^{2} \cos \left (f x +e \right ) c^{2} d}{4 f}-\frac {9 a^{2} \cos \left (f x +e \right ) c \,d^{2}}{2 f}-\frac {11 a^{2} \cos \left (f x +e \right ) d^{3}}{8 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right )}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{2} c \,d^{2}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} d^{3}}{16 f}+\frac {a^{2} d \cos \left (3 f x +3 e \right ) c^{2}}{4 f}+\frac {a^{2} d^{2} \cos \left (3 f x +3 e \right ) c}{2 f}+\frac {3 a^{2} d^{3} \cos \left (3 f x +3 e \right )}{16 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} c^{3}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c^{2} d}{2 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c \,d^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} d^{3}}{2 f}\) | \(315\) |
derivativedivides | \(\frac {a^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} c^{3} \cos \left (f x +e \right )+6 a^{2} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{2} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} c^{3} \left (f x +e \right )-3 a^{2} c^{2} d \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(329\) |
default | \(\frac {a^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} c^{3} \cos \left (f x +e \right )+6 a^{2} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{2} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} c^{3} \left (f x +e \right )-3 a^{2} c^{2} d \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(329\) |
norman | \(\frac {\left (\frac {3}{2} a^{2} c^{3}+3 a^{2} c^{2} d +\frac {21}{8} a^{2} c \,d^{2}+\frac {3}{4} a^{2} d^{3}\right ) x +\left (15 a^{2} c^{3}+30 a^{2} c^{2} d +\frac {105}{4} a^{2} c \,d^{2}+\frac {15}{2} a^{2} d^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (15 a^{2} c^{3}+30 a^{2} c^{2} d +\frac {105}{4} a^{2} c \,d^{2}+\frac {15}{2} a^{2} d^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {3}{2} a^{2} c^{3}+3 a^{2} c^{2} d +\frac {21}{8} a^{2} c \,d^{2}+\frac {3}{4} a^{2} d^{3}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} a^{2} c^{3}+15 a^{2} c^{2} d +\frac {105}{8} a^{2} c \,d^{2}+\frac {15}{4} a^{2} d^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} a^{2} c^{3}+15 a^{2} c^{2} d +\frac {105}{8} a^{2} c \,d^{2}+\frac {15}{4} a^{2} d^{3}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {20 a^{2} c^{3}+50 a^{2} c^{2} d +40 a^{2} c \,d^{2}+12 a^{2} d^{3}}{5 f}-\frac {\left (4 a^{2} c^{3}+6 a^{2} c^{2} d \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (8 a^{2} c^{3}+18 a^{2} c^{2} d +12 a^{2} c \,d^{2}+2 a^{2} d^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (12 a^{2} c^{3}+32 a^{2} c^{2} d +28 a^{2} c \,d^{2}+10 a^{2} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (16 a^{2} c^{3}+44 a^{2} c^{2} d +40 a^{2} c \,d^{2}+12 a^{2} d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2} \left (4 c^{3}+24 c^{2} d +21 c \,d^{2}+6 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 c^{3}+24 c^{2} d +21 c \,d^{2}+6 d^{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} \left (4 c^{3}+24 c^{2} d +33 c \,d^{2}+14 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} \left (4 c^{3}+24 c^{2} d +33 c \,d^{2}+14 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\) | \(688\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=-\frac {8 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 40 \, {\left (a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} f x + 80 \, {\left (a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right ) - 5 \, {\left (2 \, {\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 27 \, a^{2} c d^{2} + 10 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (218) = 436\).
Time = 0.36 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.39 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\begin {cases} \frac {a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x - \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c^{3} \cos {\left (e + f x \right )}}{f} + 3 a^{2} c^{2} d x \sin ^{2}{\left (e + f x \right )} + 3 a^{2} c^{2} d x \cos ^{2}{\left (e + f x \right )} - \frac {3 a^{2} c^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} c^{2} d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \cos {\left (e + f x \right )}}{f} + \frac {9 a^{2} c d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{2} c d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {15 a^{2} c d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {6 a^{2} c d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {9 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {4 a^{2} c d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a^{2} d^{3} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 a^{2} d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{2} d^{3} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {a^{2} d^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} d^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {4 a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {8 a^{2} d^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{2} d^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right )^{3} \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.48 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} + 480 \, {\left (f x + e\right )} a^{2} c^{3} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{3} - 960 \, a^{2} c^{3} \cos \left (f x + e\right ) - 1440 \, a^{2} c^{2} d \cos \left (f x + e\right )}{480 \, f} \]
[In]
[Out]
none
Time = 0.39 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.47 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=-\frac {a^{2} d^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {a^{2} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, a^{2} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} x + \frac {1}{2} \, {\left (2 \, a^{2} c^{3} + 3 \, a^{2} c d^{2}\right )} x + \frac {{\left (12 \, a^{2} c^{2} d + 24 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, a^{2} c^{3} + 18 \, a^{2} c^{2} d + 36 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (4 \, a^{2} c^{2} d + a^{2} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac {{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (a^{2} c^{3} + 6 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
[In]
[Out]
Time = 9.76 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.84 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx=\frac {3\,a^2\,\mathrm {atan}\left (\frac {3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,\left (3\,a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,f}-\frac {3\,a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (4\,c^3+8\,c^2\,d+7\,c\,d^2+2\,d^3\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,a^2\,c^3+6\,d\,a^2\,c^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (16\,a^2\,c^3+36\,a^2\,c^2\,d+24\,a^2\,c\,d^2+4\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (16\,a^2\,c^3+44\,a^2\,c^2\,d+40\,a^2\,c\,d^2+12\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (24\,a^2\,c^3+64\,a^2\,c^2\,d+56\,a^2\,c\,d^2+20\,a^2\,d^3\right )+4\,a^2\,c^3+\frac {12\,a^2\,d^3}{5}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+8\,a^2\,c\,d^2+10\,a^2\,c^2\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
[In]
[Out]