Integrand size = 26, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {45}{128} a^3 c^5 x+\frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f} \]
45/128*a^3*c^5*x+9/56*a^3*c^5*cos(f*x+e)^7/f+45/128*a^3*c^5*cos(f*x+e)*sin (f*x+e)/f+15/64*a^3*c^5*cos(f*x+e)^3*sin(f*x+e)/f+3/16*a^3*c^5*cos(f*x+e)^ 5*sin(f*x+e)/f+1/8*a^3*cos(f*x+e)^7*(c^5-c^5*sin(f*x+e))/f
Time = 8.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {a^3 c^5 (2520 e+2520 f x+1120 \cos (e+f x)+672 \cos (3 (e+f x))+224 \cos (5 (e+f x))+32 \cos (7 (e+f x))+1792 \sin (2 (e+f x))+280 \sin (4 (e+f x))-7 \sin (8 (e+f x)))}{7168 f} \]
(a^3*c^5*(2520*e + 2520*f*x + 1120*Cos[e + f*x] + 672*Cos[3*(e + f*x)] + 2 24*Cos[5*(e + f*x)] + 32*Cos[7*(e + f*x)] + 1792*Sin[2*(e + f*x)] + 280*Si n[4*(e + f*x)] - 7*Sin[8*(e + f*x)]))/(7168*f)
Time = 0.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3215, 3042, 3157, 3042, 3148, 3042, 3115, 3042, 3115, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \cos ^6(e+f x) (c-c \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \cos (e+f x)^6 (c-c \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \int \cos ^6(e+f x) (c-c \sin (e+f x))dx+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \int \cos (e+f x)^6 (c-c \sin (e+f x))dx+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \int \cos ^6(e+f x)dx+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \int \sin \left (e+f x+\frac {\pi }{2}\right )^6dx+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \int \cos ^4(e+f x)dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \int \sin \left (e+f x+\frac {\pi }{2}\right )^4dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(e+f x)dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^3 c^3 \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}+\frac {9}{8} c \left (\frac {c \cos ^7(e+f x)}{7 f}+c \left (\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5}{6} \left (\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3}{4} \left (\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}\right )\right )\right )\right )\right )\) |
a^3*c^3*((Cos[e + f*x]^7*(c^2 - c^2*Sin[e + f*x]))/(8*f) + (9*c*((c*Cos[e + f*x]^7)/(7*f) + c*((Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (5*((Cos[e + f* x]^3*Sin[e + f*x])/(4*f) + (3*(x/2 + (Cos[e + f*x]*Sin[e + f*x])/(2*f)))/4 ))/6)))/8)
3.3.48.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 3.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63
| method | result | size |
| parallelrisch | \(-\frac {c^{5} a^{3} \left (-2520 f x +7 \sin \left (8 f x +8 e \right )-32 \cos \left (7 f x +7 e \right )-224 \cos \left (5 f x +5 e \right )-280 \sin \left (4 f x +4 e \right )-672 \cos \left (3 f x +3 e \right )-1792 \sin \left (2 f x +2 e \right )-1120 \cos \left (f x +e \right )-2048\right )}{7168 f}\) | \(92\) |
| risch | \(\frac {45 a^{3} c^{5} x}{128}+\frac {5 c^{5} a^{3} \cos \left (f x +e \right )}{32 f}-\frac {c^{5} a^{3} \sin \left (8 f x +8 e \right )}{1024 f}+\frac {c^{5} a^{3} \cos \left (7 f x +7 e \right )}{224 f}+\frac {c^{5} a^{3} \cos \left (5 f x +5 e \right )}{32 f}+\frac {5 c^{5} a^{3} \sin \left (4 f x +4 e \right )}{128 f}+\frac {3 c^{5} a^{3} \cos \left (3 f x +3 e \right )}{32 f}+\frac {c^{5} a^{3} \sin \left (2 f x +2 e \right )}{4 f}\) | \(148\) |
| derivativedivides | \(\frac {-c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 a^{3} c^{5} \cos \left (f x +e \right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(276\) |
| default | \(\frac {-c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 a^{3} c^{5} \cos \left (f x +e \right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(276\) |
| parts | \(a^{3} c^{5} x +\frac {2 c^{5} a^{3} \cos \left (f x +e \right )}{f}-\frac {2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{f}+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}-\frac {c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\) | \(289\) |
| norman | \(\frac {\frac {4 c^{5} a^{3}}{7 f}+\frac {83 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}+\frac {45 a^{3} c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{3} c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {315 a^{3} c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {1575 a^{3} c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64}+\frac {315 a^{3} c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{3} c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {45 a^{3} c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {45 a^{3} c^{5} x \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{128}+\frac {4 c^{5} a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 c^{5} a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 f}+\frac {12 c^{5} a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {12 c^{5} a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {20 c^{5} a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {20 c^{5} a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 c^{5} a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {295 c^{5} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {3 c^{5} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {815 c^{5} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {815 c^{5} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {3 c^{5} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {295 c^{5} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {83 c^{5} a^{3} \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {45 a^{3} c^{5} x}{128}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{8}}\) | \(526\) |
-1/7168*c^5*a^3*(-2520*f*x+7*sin(8*f*x+8*e)-32*cos(7*f*x+7*e)-224*cos(5*f* x+5*e)-280*sin(4*f*x+4*e)-672*cos(3*f*x+3*e)-1792*sin(2*f*x+2*e)-1120*cos( f*x+e)-2048)/f
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {256 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} + 315 \, a^{3} c^{5} f x - 7 \, {\left (16 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} - 24 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 30 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 45 \, a^{3} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{896 \, f} \]
1/896*(256*a^3*c^5*cos(f*x + e)^7 + 315*a^3*c^5*f*x - 7*(16*a^3*c^5*cos(f* x + e)^7 - 24*a^3*c^5*cos(f*x + e)^5 - 30*a^3*c^5*cos(f*x + e)^3 - 45*a^3* c^5*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (139) = 278\).
Time = 0.75 (sec) , antiderivative size = 740, normalized size of antiderivative = 5.10 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\begin {cases} - \frac {35 a^{3} c^{5} x \sin ^{8}{\left (e + f x \right )}}{128} - \frac {35 a^{3} c^{5} x \sin ^{6}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{32} + \frac {5 a^{3} c^{5} x \sin ^{6}{\left (e + f x \right )}}{8} - \frac {105 a^{3} c^{5} x \sin ^{4}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{64} + \frac {15 a^{3} c^{5} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{8} - \frac {35 a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{6}{\left (e + f x \right )}}{32} + \frac {15 a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{8} - a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} - \frac {35 a^{3} c^{5} x \cos ^{8}{\left (e + f x \right )}}{128} + \frac {5 a^{3} c^{5} x \cos ^{6}{\left (e + f x \right )}}{8} - a^{3} c^{5} x \cos ^{2}{\left (e + f x \right )} + a^{3} c^{5} x + \frac {93 a^{3} c^{5} \sin ^{7}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{128 f} - \frac {2 a^{3} c^{5} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {511 a^{3} c^{5} \sin ^{5}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{384 f} - \frac {11 a^{3} c^{5} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 a^{3} c^{5} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {6 a^{3} c^{5} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {385 a^{3} c^{5} \sin ^{3}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{384 f} - \frac {5 a^{3} c^{5} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {16 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {8 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {6 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {35 a^{3} c^{5} \sin {\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{128 f} - \frac {5 a^{3} c^{5} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{8 f} + \frac {a^{3} c^{5} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {32 a^{3} c^{5} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {16 a^{3} c^{5} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {4 a^{3} c^{5} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {2 a^{3} c^{5} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{3} \left (- c \sin {\left (e \right )} + c\right )^{5} & \text {otherwise} \end {cases} \]
Piecewise((-35*a**3*c**5*x*sin(e + f*x)**8/128 - 35*a**3*c**5*x*sin(e + f* x)**6*cos(e + f*x)**2/32 + 5*a**3*c**5*x*sin(e + f*x)**6/8 - 105*a**3*c**5 *x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*a**3*c**5*x*sin(e + f*x)**4*cos (e + f*x)**2/8 - 35*a**3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15*a* *3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/8 - a**3*c**5*x*sin(e + f*x)**2 - 35*a**3*c**5*x*cos(e + f*x)**8/128 + 5*a**3*c**5*x*cos(e + f*x)**6/8 - a **3*c**5*x*cos(e + f*x)**2 + a**3*c**5*x + 93*a**3*c**5*sin(e + f*x)**7*co s(e + f*x)/(128*f) - 2*a**3*c**5*sin(e + f*x)**6*cos(e + f*x)/f + 511*a**3 *c**5*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) - 11*a**3*c**5*sin(e + f*x)* *5*cos(e + f*x)/(8*f) - 4*a**3*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f + 6* a**3*c**5*sin(e + f*x)**4*cos(e + f*x)/f + 385*a**3*c**5*sin(e + f*x)**3*c os(e + f*x)**5/(384*f) - 5*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(3*f) - 16*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 8*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**3/f - 6*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)/f + 35*a**3*c**5*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*a**3*c**5*sin(e + f *x)*cos(e + f*x)**5/(8*f) + a**3*c**5*sin(e + f*x)*cos(e + f*x)/f - 32*a** 3*c**5*cos(e + f*x)**7/(35*f) + 16*a**3*c**5*cos(e + f*x)**5/(5*f) - 4*a** 3*c**5*cos(e + f*x)**3/f + 2*a**3*c**5*cos(e + f*x)/f, Ne(f, 0)), (x*(a*si n(e) + a)**3*(-c*sin(e) + c)**5, True))
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (134) = 268\).
Time = 0.21 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.94 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {6144 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} + 43008 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} + 215040 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} - 35 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 1120 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} - 53760 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 107520 \, {\left (f x + e\right )} a^{3} c^{5} + 215040 \, a^{3} c^{5} \cos \left (f x + e\right )}{107520 \, f} \]
1/107520*(6144*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*a^3*c^5 + 43008*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^5 + 215040*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^ 5 - 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 16 8*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*a^3*c^5 + 1120*(4*sin(2*f*x + 2 *e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^3*c^5 - 53760*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^5 + 107520*(f*x + e)*a^3*c^ 5 + 215040*a^3*c^5*cos(f*x + e))/f
Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {45}{128} \, a^{3} c^{5} x + \frac {a^{3} c^{5} \cos \left (7 \, f x + 7 \, e\right )}{224 \, f} + \frac {a^{3} c^{5} \cos \left (5 \, f x + 5 \, e\right )}{32 \, f} + \frac {3 \, a^{3} c^{5} \cos \left (3 \, f x + 3 \, e\right )}{32 \, f} + \frac {5 \, a^{3} c^{5} \cos \left (f x + e\right )}{32 \, f} - \frac {a^{3} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {5 \, a^{3} c^{5} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {a^{3} c^{5} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
45/128*a^3*c^5*x + 1/224*a^3*c^5*cos(7*f*x + 7*e)/f + 1/32*a^3*c^5*cos(5*f *x + 5*e)/f + 3/32*a^3*c^5*cos(3*f*x + 3*e)/f + 5/32*a^3*c^5*cos(f*x + e)/ f - 1/1024*a^3*c^5*sin(8*f*x + 8*e)/f + 5/128*a^3*c^5*sin(4*f*x + 4*e)/f + 1/4*a^3*c^5*sin(2*f*x + 2*e)/f
Time = 9.00 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.57 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {a^3\,c^5\,\left (\frac {315\,e}{2}+581\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+256\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2065\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+5376\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+5376\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+5705\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+8960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-5705\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+8960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+1792\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-2065\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}+1792\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-581\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}+\frac {315\,f\,x}{2}+1260\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+4410\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+8820\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+11025\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+8820\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+4410\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )+1260\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (e+f\,x\right )+\frac {315\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}\,\left (e+f\,x\right )}{2}+256\right )}{448\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^8} \]
(a^3*c^5*((315*e)/2 + 581*tan(e/2 + (f*x)/2) + 256*tan(e/2 + (f*x)/2)^2 + 2065*tan(e/2 + (f*x)/2)^3 + 5376*tan(e/2 + (f*x)/2)^4 + 21*tan(e/2 + (f*x) /2)^5 + 5376*tan(e/2 + (f*x)/2)^6 + 5705*tan(e/2 + (f*x)/2)^7 + 8960*tan(e /2 + (f*x)/2)^8 - 5705*tan(e/2 + (f*x)/2)^9 + 8960*tan(e/2 + (f*x)/2)^10 - 21*tan(e/2 + (f*x)/2)^11 + 1792*tan(e/2 + (f*x)/2)^12 - 2065*tan(e/2 + (f *x)/2)^13 + 1792*tan(e/2 + (f*x)/2)^14 - 581*tan(e/2 + (f*x)/2)^15 + (315* f*x)/2 + 1260*tan(e/2 + (f*x)/2)^2*(e + f*x) + 4410*tan(e/2 + (f*x)/2)^4*( e + f*x) + 8820*tan(e/2 + (f*x)/2)^6*(e + f*x) + 11025*tan(e/2 + (f*x)/2)^ 8*(e + f*x) + 8820*tan(e/2 + (f*x)/2)^10*(e + f*x) + 4410*tan(e/2 + (f*x)/ 2)^12*(e + f*x) + 1260*tan(e/2 + (f*x)/2)^14*(e + f*x) + (315*tan(e/2 + (f *x)/2)^16*(e + f*x))/2 + 256))/(448*f*(tan(e/2 + (f*x)/2)^2 + 1)^8)