\(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx\) [247]

Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\frac {55}{128} a^3 c^6 x+\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f} \]

[Out]

55/128*a^3*c^6*x+11/56*a^3*c^6*cos(f*x+e)^7/f+55/128*a^3*c^6*cos(f*x+e)*sin(f*x+e)/f+55/192*a^3*c^6*cos(f*x+e)
^3*sin(f*x+e)/f+11/48*a^3*c^6*cos(f*x+e)^5*sin(f*x+e)/f+1/9*a^3*cos(f*x+e)^7*(c^3-c^3*sin(f*x+e))^2/f+11/72*a^
3*cos(f*x+e)^7*(c^6-c^6*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2757, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac {55}{128} a^3 c^6 x+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]

[Out]

(55*a^3*c^6*x)/128 + (11*a^3*c^6*Cos[e + f*x]^7)/(56*f) + (55*a^3*c^6*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (55
*a^3*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (11*a^3*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) + (a^3*Cos[e +
f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(9*f) + (11*a^3*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(72*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[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])

Rule 2757

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] + Dist[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] && IntegersQ[2*m, 2*p]

Rule 2815

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[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] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3 \, dx \\ & = \frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {1}{9} \left (11 a^3 c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx \\ & = \frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{8} \left (11 a^3 c^5\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{8} \left (11 a^3 c^6\right ) \int \cos ^6(e+f x) \, dx \\ & = \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{48} \left (55 a^3 c^6\right ) \int \cos ^4(e+f x) \, dx \\ & = \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{64} \left (55 a^3 c^6\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{128} \left (55 a^3 c^6\right ) \int 1 \, dx \\ & = \frac {55}{128} a^3 c^6 x+\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\frac {a^3 c^6 (27720 e+27720 f x+16632 \cos (e+f x)+9744 \cos (3 (e+f x))+3024 \cos (5 (e+f x))+324 \cos (7 (e+f x))-28 \cos (9 (e+f x))+18144 \sin (2 (e+f x))+1512 \sin (4 (e+f x))-672 \sin (6 (e+f x))-189 \sin (8 (e+f x)))}{64512 f} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]

[Out]

(a^3*c^6*(27720*e + 27720*f*x + 16632*Cos[e + f*x] + 9744*Cos[3*(e + f*x)] + 3024*Cos[5*(e + f*x)] + 324*Cos[7
*(e + f*x)] - 28*Cos[9*(e + f*x)] + 18144*Sin[2*(e + f*x)] + 1512*Sin[4*(e + f*x)] - 672*Sin[6*(e + f*x)] - 18
9*Sin[8*(e + f*x)]))/(64512*f)

Maple [A] (verified)

Time = 3.69 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.62

method	result	size
parallelrisch	\(-\frac {\left (-990 f x +\cos \left (9 f x +9 e \right )-648 \sin \left (2 f x +2 e \right )-54 \sin \left (4 f x +4 e \right )+24 \sin \left (6 f x +6 e \right )+\frac {27 \sin \left (8 f x +8 e \right )}{4}-594 \cos \left (f x +e \right )-348 \cos \left (3 f x +3 e \right )-108 \cos \left (5 f x +5 e \right )-\frac {81 \cos \left (7 f x +7 e \right )}{7}-\frac {7424}{7}\right ) a^{3} c^{6}}{2304 f}\)	\(112\)
risch	\(\frac {55 a^{3} c^{6} x}{128}+\frac {33 c^{6} a^{3} \cos \left (f x +e \right )}{128 f}-\frac {c^{6} a^{3} \cos \left (9 f x +9 e \right )}{2304 f}-\frac {3 c^{6} a^{3} \sin \left (8 f x +8 e \right )}{1024 f}+\frac {9 c^{6} a^{3} \cos \left (7 f x +7 e \right )}{1792 f}-\frac {c^{6} a^{3} \sin \left (6 f x +6 e \right )}{96 f}+\frac {3 c^{6} a^{3} \cos \left (5 f x +5 e \right )}{64 f}+\frac {3 c^{6} a^{3} \sin \left (4 f x +4 e \right )}{128 f}+\frac {29 c^{6} a^{3} \cos \left (3 f x +3 e \right )}{192 f}+\frac {9 c^{6} a^{3} \sin \left (2 f x +2 e \right )}{32 f}\)	\(188\)
derivativedivides	\(\frac {c^{6} a^{3} \left (f x +e \right )+3 c^{6} a^{3} \cos \left (f x +e \right )-\frac {8 c^{6} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-6 c^{6} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {6 c^{6} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+8 c^{6} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {c^{6} a^{3} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{9}-3 c^{6} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\)	\(297\)
default	\(\frac {c^{6} a^{3} \left (f x +e \right )+3 c^{6} a^{3} \cos \left (f x +e \right )-\frac {8 c^{6} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-6 c^{6} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {6 c^{6} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+8 c^{6} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {c^{6} a^{3} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{9}-3 c^{6} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\)	\(297\)
parts	\(a^{3} c^{6} x -\frac {c^{6} a^{3} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{9 f}+\frac {3 c^{6} a^{3} \cos \left (f x +e \right )}{f}-\frac {8 c^{6} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {6 c^{6} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {6 c^{6} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {8 c^{6} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {3 c^{6} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\)	\(310\)
norman	\(\frac {\frac {73 c^{6} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}+\frac {58 c^{6} a^{3}}{63 f}+\frac {55 a^{3} c^{6} x \left (\tan ^{18}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{128}+\frac {48 c^{6} a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {80 c^{6} a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {88 c^{6} a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {16 c^{6} a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {16 c^{6} a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 f}+\frac {120 c^{6} a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 f}+\frac {949 c^{6} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 f}-\frac {17 c^{6} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}+\frac {699 c^{6} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {699 c^{6} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}+\frac {17 c^{6} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}+\frac {36 c^{6} a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {6 c^{6} a^{3} \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {949 c^{6} a^{3} \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{96 f}-\frac {73 c^{6} a^{3} \left (\tan ^{17}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {495 a^{3} c^{6} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{128}+\frac {495 a^{3} c^{6} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {1155 a^{3} c^{6} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {3465 a^{3} c^{6} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64}+\frac {3465 a^{3} c^{6} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64}+\frac {1155 a^{3} c^{6} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {495 a^{3} c^{6} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {495 a^{3} c^{6} x \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{128}+\frac {55 a^{3} c^{6} x}{128}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{9}}\)	\(568\)

[In]

int((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

-1/2304*(-990*f*x+cos(9*f*x+9*e)-648*sin(2*f*x+2*e)-54*sin(4*f*x+4*e)+24*sin(6*f*x+6*e)+27/4*sin(8*f*x+8*e)-59
4*cos(f*x+e)-348*cos(3*f*x+3*e)-108*cos(5*f*x+5*e)-81/7*cos(7*f*x+7*e)-7424/7)*a^3*c^6/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=-\frac {896 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} - 4608 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \, a^{3} c^{6} f x + 21 \, {\left (144 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \, a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \, a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \, a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8064 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

-1/8064*(896*a^3*c^6*cos(f*x + e)^9 - 4608*a^3*c^6*cos(f*x + e)^7 - 3465*a^3*c^6*f*x + 21*(144*a^3*c^6*cos(f*x
+ e)^7 - 88*a^3*c^6*cos(f*x + e)^5 - 110*a^3*c^6*cos(f*x + e)^3 - 165*a^3*c^6*cos(f*x + e))*sin(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (172) = 344\).

Time = 1.04 (sec) , antiderivative size = 838, normalized size of antiderivative = 4.66 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\begin {cases} - \frac {105 a^{3} c^{6} x \sin ^{8}{\left (e + f x \right )}}{128} - \frac {105 a^{3} c^{6} x \sin ^{6}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{32} + \frac {5 a^{3} c^{6} x \sin ^{6}{\left (e + f x \right )}}{2} - \frac {315 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{64} + \frac {15 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )}}{4} - \frac {105 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{6}{\left (e + f x \right )}}{32} + \frac {15 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {105 a^{3} c^{6} x \cos ^{8}{\left (e + f x \right )}}{128} + \frac {5 a^{3} c^{6} x \cos ^{6}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \cos ^{4}{\left (e + f x \right )}}{4} + a^{3} c^{6} x - \frac {a^{3} c^{6} \sin ^{8}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {279 a^{3} c^{6} \sin ^{7}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{128 f} - \frac {8 a^{3} c^{6} \sin ^{6}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {511 a^{3} c^{6} \sin ^{5}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{128 f} - \frac {11 a^{3} c^{6} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {16 a^{3} c^{6} \sin ^{4}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {6 a^{3} c^{6} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {385 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{128 f} - \frac {20 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {15 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {64 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {8 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {105 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{128 f} - \frac {5 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{2 f} + \frac {9 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {128 a^{3} c^{6} \cos ^{9}{\left (e + f x \right )}}{315 f} + \frac {16 a^{3} c^{6} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {16 a^{3} c^{6} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 a^{3} c^{6} \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 )^{6} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))**3*(c-c*sin(f*x+e))**6,x)

[Out]

Piecewise((-105*a**3*c**6*x*sin(e + f*x)**8/128 - 105*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*a**3*
c**6*x*sin(e + f*x)**6/2 - 315*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*a**3*c**6*x*sin(e + f*x)**4
*cos(e + f*x)**2/2 - 9*a**3*c**6*x*sin(e + f*x)**4/4 - 105*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15
*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**4/2 - 9*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**2/2 - 105*a**3*c*
*6*x*cos(e + f*x)**8/128 + 5*a**3*c**6*x*cos(e + f*x)**6/2 - 9*a**3*c**6*x*cos(e + f*x)**4/4 + a**3*c**6*x - a
**3*c**6*sin(e + f*x)**8*cos(e + f*x)/f + 279*a**3*c**6*sin(e + f*x)**7*cos(e + f*x)/(128*f) - 8*a**3*c**6*sin
(e + f*x)**6*cos(e + f*x)**3/(3*f) + 511*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)**3/(128*f) - 11*a**3*c**6*sin(
e + f*x)**5*cos(e + f*x)/(2*f) - 16*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)**5/(5*f) + 6*a**3*c**6*sin(e + f*x)
**4*cos(e + f*x)/f + 385*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**5/(128*f) - 20*a**3*c**6*sin(e + f*x)**3*cos(
e + f*x)**3/(3*f) + 15*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 64*a**3*c**6*sin(e + f*x)**2*cos(e + f*x
)**7/(35*f) + 8*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f - 8*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)/f + 105
*a**3*c**6*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*a**3*c**6*sin(e + f*x)*cos(e + f*x)**5/(2*f) + 9*a**3*c**6
*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 128*a**3*c**6*cos(e + f*x)**9/(315*f) + 16*a**3*c**6*cos(e + f*x)**5/(5*
f) - 16*a**3*c**6*cos(e + f*x)**3/(3*f) + 3*a**3*c**6*cos(e + f*x)/f, Ne(f, 0)), (x*(a*sin(e) + a)**3*(-c*sin(
e) + c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.67 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=-\frac {1024 \, {\left (35 \, \cos \left (f x + e\right )^{9} - 180 \, \cos \left (f x + e\right )^{7} + 378 \, \cos \left (f x + e\right )^{5} - 420 \, \cos \left (f x + e\right )^{3} + 315 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 129024 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 860160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} + 315 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 13440 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} + 60480 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 322560 \, {\left (f x + e\right )} a^{3} c^{6} - 967680 \, a^{3} c^{6} \cos \left (f x + e\right )}{322560 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-1/322560*(1024*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 420*cos(f*x + e)^3 + 315*cos(f*
x + e))*a^3*c^6 - 129024*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^6 - 860160*(cos(f*x +
e)^3 - 3*cos(f*x + e))*a^3*c^6 + 315*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 168*sin(
4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*a^3*c^6 - 13440*(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^6 + 60480*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c^6 - 322
560*(f*x + e)*a^3*c^6 - 967680*a^3*c^6*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.04 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\frac {55}{128} \, a^{3} c^{6} x - \frac {a^{3} c^{6} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac {9 \, a^{3} c^{6} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac {3 \, a^{3} c^{6} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {29 \, a^{3} c^{6} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac {33 \, a^{3} c^{6} \cos \left (f x + e\right )}{128 \, f} - \frac {3 \, a^{3} c^{6} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} - \frac {a^{3} c^{6} \sin \left (6 \, f x + 6 \, e\right )}{96 \, f} + \frac {3 \, a^{3} c^{6} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {9 \, a^{3} c^{6} \sin \left (2 \, f x + 2 \, e\right )}{32 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

55/128*a^3*c^6*x - 1/2304*a^3*c^6*cos(9*f*x + 9*e)/f + 9/1792*a^3*c^6*cos(7*f*x + 7*e)/f + 3/64*a^3*c^6*cos(5*
f*x + 5*e)/f + 29/192*a^3*c^6*cos(3*f*x + 3*e)/f + 33/128*a^3*c^6*cos(f*x + e)/f - 3/1024*a^3*c^6*sin(8*f*x +
8*e)/f - 1/96*a^3*c^6*sin(6*f*x + 6*e)/f + 3/128*a^3*c^6*sin(4*f*x + 4*e)/f + 9/32*a^3*c^6*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.24 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx=\frac {a^3\,c^6\,\left (3465\,e+9198\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+18432\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+79716\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+138240\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4284\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+387072\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+176148\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+290304\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+645120\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-176148\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+236544\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+4284\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}+129024\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-79716\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}+48384\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-9198\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{17}+3465\,f\,x+31185\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+124740\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+291060\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+436590\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+436590\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+291060\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )+124740\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (e+f\,x\right )+31185\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}\,\left (e+f\,x\right )+3465\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{18}\,\left (e+f\,x\right )+7424\right )}{8064\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^9} \]

[In]

int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6,x)

[Out]

(a^3*c^6*(3465*e + 9198*tan(e/2 + (f*x)/2) + 18432*tan(e/2 + (f*x)/2)^2 + 79716*tan(e/2 + (f*x)/2)^3 + 138240*
tan(e/2 + (f*x)/2)^4 - 4284*tan(e/2 + (f*x)/2)^5 + 387072*tan(e/2 + (f*x)/2)^6 + 176148*tan(e/2 + (f*x)/2)^7 +
290304*tan(e/2 + (f*x)/2)^8 + 645120*tan(e/2 + (f*x)/2)^10 - 176148*tan(e/2 + (f*x)/2)^11 + 236544*tan(e/2 +
(f*x)/2)^12 + 4284*tan(e/2 + (f*x)/2)^13 + 129024*tan(e/2 + (f*x)/2)^14 - 79716*tan(e/2 + (f*x)/2)^15 + 48384*
tan(e/2 + (f*x)/2)^16 - 9198*tan(e/2 + (f*x)/2)^17 + 3465*f*x + 31185*tan(e/2 + (f*x)/2)^2*(e + f*x) + 124740*
tan(e/2 + (f*x)/2)^4*(e + f*x) + 291060*tan(e/2 + (f*x)/2)^6*(e + f*x) + 436590*tan(e/2 + (f*x)/2)^8*(e + f*x)
+ 436590*tan(e/2 + (f*x)/2)^10*(e + f*x) + 291060*tan(e/2 + (f*x)/2)^12*(e + f*x) + 124740*tan(e/2 + (f*x)/2)
^14*(e + f*x) + 31185*tan(e/2 + (f*x)/2)^16*(e + f*x) + 3465*tan(e/2 + (f*x)/2)^18*(e + f*x) + 7424))/(8064*f*
(tan(e/2 + (f*x)/2)^2 + 1)^9)