Optimal. Leaf size=164 \[ \frac {a^3 \left (c^2+6 c d+5 d^2\right ) \cos ^3(e+f x)}{3 f}-\frac {a^3 \left (12 c^2+30 c d+13 d^2\right ) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^3 x \left (20 c^2+30 c d+13 d^2\right )-\frac {4 a^3 (c+d)^2 \cos (e+f x)}{f}-\frac {a^3 d (2 c+3 d) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac {a^3 d^2 \cos ^5(e+f x)}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 189, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2761, 2751, 2645, 2638, 2635, 8, 2633} \[ \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos ^3(e+f x)}{60 f}-\frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{5 f}-\frac {3 a^3 \left (20 c^2+30 c d+13 d^2\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {1}{8} a^3 x \left (20 c^2+30 c d+13 d^2\right )-\frac {d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2645
Rule 2751
Rule 2761
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx &=-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {\int (a+a \sin (e+f x))^3 \left (a \left (5 c^2+4 d^2\right )+a (10 c-d) d \sin (e+f x)\right ) \, dx}{5 a}\\ &=-\frac {(10 c-d) d \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} \left (20 c^2+30 c d+13 d^2\right ) \int (a+a \sin (e+f x))^3 \, dx\\ &=-\frac {(10 c-d) d \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} \left (20 c^2+30 c d+13 d^2\right ) \int \left (a^3+3 a^3 \sin (e+f x)+3 a^3 \sin ^2(e+f x)+a^3 \sin ^3(e+f x)\right ) \, dx\\ &=\frac {1}{20} a^3 \left (20 c^2+30 c d+13 d^2\right ) x-\frac {(10 c-d) d \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} \left (a^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \int \sin ^3(e+f x) \, dx+\frac {1}{20} \left (3 a^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \int \sin (e+f x) \, dx+\frac {1}{20} \left (3 a^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \int \sin ^2(e+f x) \, dx\\ &=\frac {1}{20} a^3 \left (20 c^2+30 c d+13 d^2\right ) x-\frac {3 a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{20 f}-\frac {3 a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(10 c-d) d \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{40} \left (3 a^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \int 1 \, dx-\frac {\left (a^3 \left (20 c^2+30 c d+13 d^2\right )\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{20 f}\\ &=\frac {1}{8} a^3 \left (20 c^2+30 c d+13 d^2\right ) x-\frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{5 f}+\frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos ^3(e+f x)}{60 f}-\frac {3 a^3 \left (20 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(10 c-d) d \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.74, size = 177, normalized size = 1.08 \[ -\frac {a^3 \cos (e+f x) \left (30 \left (20 c^2+30 c d+13 d^2\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (8 \left (5 c^2+30 c d+19 d^2\right ) \sin ^2(e+f x)+15 \left (12 c^2+30 c d+13 d^2\right ) \sin (e+f x)+8 \left (55 c^2+90 c d+38 d^2\right )+30 d (2 c+3 d) \sin ^3(e+f x)+24 d^2 \sin ^4(e+f x)\right )\right )}{120 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 180, normalized size = 1.10 \[ -\frac {24 \, a^{3} d^{2} \cos \left (f x + e\right )^{5} - 40 \, {\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} f x + 480 \, {\left (a^{3} c^{2} + 2 \, a^{3} c d + a^{3} d^{2}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (12 \, a^{3} c^{2} + 34 \, a^{3} c d + 19 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 251, normalized size = 1.53 \[ -\frac {a^{3} d^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {2 \, a^{3} c d \cos \left (f x + e\right )}{f} - \frac {a^{3} d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {3}{8} \, {\left (4 \, a^{3} c^{2} + 10 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} x + \frac {1}{2} \, {\left (2 \, a^{3} c^{2} + a^{3} d^{2}\right )} x + \frac {{\left (4 \, a^{3} c^{2} + 24 \, a^{3} c d + 17 \, a^{3} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, a^{3} c^{2} + 36 \, a^{3} c d + 23 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (3 \, a^{3} c^{2} + 8 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.32, size = 319, normalized size = 1.95 \[ \frac {-\frac {c^{2} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{3} c d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 c^{2} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 c^{2} a^{3} \cos \left (f x +e \right )+6 a^{3} c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+c^{2} a^{3} \left (f x +e \right )-2 a^{3} c d \cos \left (f x +e \right )+a^{3} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 308, normalized size = 1.88 \[ \frac {160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} + 480 \, {\left (f x + e\right )} a^{3} c^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c d + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} d^{2} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{2} + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{2} - 1440 \, a^{3} c^{2} \cos \left (f x + e\right ) - 960 \, a^{3} c d \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.36, size = 493, normalized size = 3.01 \[ \frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,\left (5\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,a^3\,c^2+19\,a^3\,c\,d+\frac {25\,a^3\,d^2}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (6\,a^3\,c^2+19\,a^3\,c\,d+\frac {25\,a^3\,d^2}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (28\,a^3\,c^2+40\,a^3\,c\,d+12\,a^3\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {92\,a^3\,c^2}{3}+56\,a^3\,c\,d+\frac {76\,a^3\,d^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {136\,a^3\,c^2}{3}+80\,a^3\,c\,d+\frac {116\,a^3\,d^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (6\,a^3\,c^2+4\,d\,a^3\,c\right )+\frac {22\,a^3\,c^2}{3}+\frac {76\,a^3\,d^2}{15}+12\,a^3\,c\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.71, size = 702, normalized size = 4.28 \[ \begin {cases} \frac {3 a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{3} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c^{2} x - \frac {a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{3} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{3} c^{2} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{3} c d x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 a^{3} c d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + 3 a^{3} c d x \sin ^{2}{\left (e + f x \right )} + \frac {3 a^{3} c d x \cos ^{4}{\left (e + f x \right )}}{4} + 3 a^{3} c d x \cos ^{2}{\left (e + f x \right )} - \frac {5 a^{3} c d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {6 a^{3} c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{3} c d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {3 a^{3} c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a^{3} c d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {2 a^{3} c d \cos {\left (e + f x \right )}}{f} + \frac {9 a^{3} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{3} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{3} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{3} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{3} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a^{3} d^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {15 a^{3} d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 a^{3} d^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{3} d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {9 a^{3} d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{3} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {8 a^{3} d^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{3} d^{2} \cos ^{3}{\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\relax (e )}\right )^{2} \left (a \sin {\relax (e )} + a\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________