\(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx\) [249]

Optimal result

Integrand size = 26, antiderivative size = 112 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {5}{16} a^3 c^4 x+\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f} \]

[Out]

5/16*a^3*c^4*x+1/7*a^3*c^4*cos(f*x+e)^7/f+5/16*a^3*c^4*cos(f*x+e)*sin(f*x+e)/f+5/24*a^3*c^4*cos(f*x+e)^3*sin(f
*x+e)/f+1/6*a^3*c^4*cos(f*x+e)^5*sin(f*x+e)/f

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {5}{16} a^3 c^4 x \]

[In]

Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]

[Out]

(5*a^3*c^4*x)/16 + (a^3*c^4*Cos[e + f*x]^7)/(7*f) + (5*a^3*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^4*
Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^4*Cos[e + f*x]^5*Sin[e + f*x])/(6*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 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)) \, dx \\ & = \frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\left (a^3 c^4\right ) \int \cos ^6(e+f x) \, dx \\ & = \frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{6} \left (5 a^3 c^4\right ) \int \cos ^4(e+f x) \, dx \\ & = \frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{8} \left (5 a^3 c^4\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{16} \left (5 a^3 c^4\right ) \int 1 \, dx \\ & = \frac {5}{16} a^3 c^4 x+\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {a^3 c^4 (420 e+420 f x+105 \cos (e+f x)+63 \cos (3 (e+f x))+21 \cos (5 (e+f x))+3 \cos (7 (e+f x))+315 \sin (2 (e+f x))+63 \sin (4 (e+f x))+7 \sin (6 (e+f x)))}{1344 f} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]

[Out]

(a^3*c^4*(420*e + 420*f*x + 105*Cos[e + f*x] + 63*Cos[3*(e + f*x)] + 21*Cos[5*(e + f*x)] + 3*Cos[7*(e + f*x)]
+ 315*Sin[2*(e + f*x)] + 63*Sin[4*(e + f*x)] + 7*Sin[6*(e + f*x)]))/(1344*f)

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\frac {c^{4} a^{3} \left (420 f x +105 \cos \left (f x +e \right )+3 \cos \left (7 f x +7 e \right )+7 \sin \left (6 f x +6 e \right )+21 \cos \left (5 f x +5 e \right )+63 \sin \left (4 f x +4 e \right )+63 \cos \left (3 f x +3 e \right )+315 \sin \left (2 f x +2 e \right )+192\right )}{1344 f}\)	\(92\)
risch	\(\frac {5 a^{3} c^{4} x}{16}+\frac {5 c^{4} a^{3} \cos \left (f x +e \right )}{64 f}+\frac {c^{4} a^{3} \cos \left (7 f x +7 e \right )}{448 f}+\frac {c^{4} a^{3} \sin \left (6 f x +6 e \right )}{192 f}+\frac {c^{4} a^{3} \cos \left (5 f x +5 e \right )}{64 f}+\frac {3 c^{4} a^{3} \sin \left (4 f x +4 e \right )}{64 f}+\frac {3 c^{4} a^{3} \cos \left (3 f x +3 e \right )}{64 f}+\frac {15 c^{4} a^{3} \sin \left (2 f x +2 e \right )}{64 f}\)	\(148\)
derivativedivides	\(\frac {-\frac {c^{4} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-c^{4} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 c^{4} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 c^{4} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-c^{4} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 c^{4} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{3} c^{4} \cos \left (f x +e \right )+c^{4} a^{3} \left (f x +e \right )}{f}\)	\(255\)
default	\(\frac {-\frac {c^{4} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-c^{4} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 c^{4} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 c^{4} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-c^{4} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 c^{4} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{3} c^{4} \cos \left (f x +e \right )+c^{4} a^{3} \left (f x +e \right )}{f}\)	\(255\)
parts	\(a^{3} c^{4} x -\frac {c^{4} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}+\frac {c^{4} a^{3} \cos \left (f x +e \right )}{f}-\frac {3 c^{4} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {c^{4} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{f}+\frac {3 c^{4} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {3 c^{4} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {c^{4} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(268\)
norman	\(\frac {\frac {2 c^{4} a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 c^{4} a^{3}}{7 f}+\frac {5 a^{3} c^{4} x}{16}+\frac {6 c^{4} a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {10 c^{4} a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {35 a^{3} c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {105 a^{3} c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {175 a^{3} c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {175 a^{3} c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {105 a^{3} c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {35 a^{3} c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {5 a^{3} c^{4} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {11 c^{4} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {7 c^{4} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}+\frac {85 c^{4} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {85 c^{4} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {7 c^{4} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}-\frac {11 c^{4} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{7}}\)	\(374\)

[In]

int((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

1/1344*c^4*a^3*(420*f*x+105*cos(f*x+e)+3*cos(7*f*x+7*e)+7*sin(6*f*x+6*e)+21*cos(5*f*x+5*e)+63*sin(4*f*x+4*e)+6
3*cos(3*f*x+3*e)+315*sin(2*f*x+2*e)+192)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {48 \, a^{3} c^{4} \cos \left (f x + e\right )^{7} + 105 \, a^{3} c^{4} f x + 7 \, {\left (8 \, a^{3} c^{4} \cos \left (f x + e\right )^{5} + 10 \, a^{3} c^{4} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

1/336*(48*a^3*c^4*cos(f*x + e)^7 + 105*a^3*c^4*f*x + 7*(8*a^3*c^4*cos(f*x + e)^5 + 10*a^3*c^4*cos(f*x + e)^3 +
15*a^3*c^4*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. 631 vs. \(2 (107) = 214\).

Time = 0.54 (sec) , antiderivative size = 631, normalized size of antiderivative = 5.63 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\begin {cases} - \frac {5 a^{3} c^{4} x \sin ^{6}{\left (e + f x \right )}}{16} - \frac {15 a^{3} c^{4} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {15 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {5 a^{3} c^{4} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {3 a^{3} c^{4} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c^{4} x - \frac {a^{3} c^{4} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {11 a^{3} c^{4} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {2 a^{3} c^{4} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a^{3} c^{4} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{3} c^{4} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {15 a^{3} c^{4} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {8 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {4 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{3} c^{4} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {9 a^{3} c^{4} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {3 a^{3} c^{4} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {16 a^{3} c^{4} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {8 a^{3} c^{4} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {2 a^{3} c^{4} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {a^{3} c^{4} \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 )^{4} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))**3*(c-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-5*a**3*c**4*x*sin(e + f*x)**6/16 - 15*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*a**3*c**4
*x*sin(e + f*x)**4/8 - 15*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*a**3*c**4*x*sin(e + f*x)**2*cos(e
+ f*x)**2/4 - 3*a**3*c**4*x*sin(e + f*x)**2/2 - 5*a**3*c**4*x*cos(e + f*x)**6/16 + 9*a**3*c**4*x*cos(e + f*x)
**4/8 - 3*a**3*c**4*x*cos(e + f*x)**2/2 + a**3*c**4*x - a**3*c**4*sin(e + f*x)**6*cos(e + f*x)/f + 11*a**3*c**
4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/f + 3*a**3*c**4*sin(e + f*
x)**4*cos(e + f*x)/f + 5*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*a**3*c**4*sin(e + f*x)**3*cos(e
+ f*x)/(8*f) - 8*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**3
/f - 3*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f + 5*a**3*c**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*a**3*c**
4*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*a**3*c**4*sin(e + f*x)*cos(e + f*x)/(2*f) - 16*a**3*c**4*cos(e + f*x)
**7/(35*f) + 8*a**3*c**4*cos(e + f*x)**5/(5*f) - 2*a**3*c**4*cos(e + f*x)**3/f + a**3*c**4*cos(e + f*x)/f, Ne(
f, 0)), (x*(a*sin(e) + a)**3*(-c*sin(e) + c)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (102) = 204\).

Time = 0.20 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.29 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 1344 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 6720 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} - 35 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} + 630 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} - 5040 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} + 6720 \, {\left (f x + e\right )} a^{3} c^{4} + 6720 \, a^{3} c^{4} \cos \left (f x + e\right )}{6720 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

1/6720*(192*(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^4 + 1344*(3*cos
(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^4 + 6720*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^4 -
35*(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^4 + 630*(12*f*x + 1
2*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c^4 - 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^4 + 6720*(f
*x + e)*a^3*c^4 + 6720*a^3*c^4*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.31 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {5}{16} \, a^{3} c^{4} x + \frac {a^{3} c^{4} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {a^{3} c^{4} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {3 \, a^{3} c^{4} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {5 \, a^{3} c^{4} \cos \left (f x + e\right )}{64 \, f} + \frac {a^{3} c^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {3 \, a^{3} c^{4} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {15 \, a^{3} c^{4} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

5/16*a^3*c^4*x + 1/448*a^3*c^4*cos(7*f*x + 7*e)/f + 1/64*a^3*c^4*cos(5*f*x + 5*e)/f + 3/64*a^3*c^4*cos(3*f*x +
3*e)/f + 5/64*a^3*c^4*cos(f*x + e)/f + 1/192*a^3*c^4*sin(6*f*x + 6*e)/f + 3/64*a^3*c^4*sin(4*f*x + 4*e)/f + 1
5/64*a^3*c^4*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 10.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.69 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (\frac {a^3\,c^4\,\left (735\,e+735\,f\,x+672\right )}{336}-\frac {35\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,c^4\,\left (2205\,e+2205\,f\,x+2016\right )}{336}-\frac {105\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^3\,c^4\,\left (3675\,e+3675\,f\,x+3360\right )}{336}-\frac {175\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+\frac {7\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6}+\frac {85\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{24}-\frac {85\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}-\frac {7\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{6}-\frac {11\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}+\frac {a^3\,c^4\,\left (105\,e+105\,f\,x+96\right )}{336}+\frac {11\,a^3\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}-\frac {5\,a^3\,c^4\,\left (e+f\,x\right )}{16}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^7}+\frac {5\,a^3\,c^4\,x}{16} \]

[In]

int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4,x)

[Out]

(tan(e/2 + (f*x)/2)^12*((a^3*c^4*(735*e + 735*f*x + 672))/336 - (35*a^3*c^4*(e + f*x))/16) + tan(e/2 + (f*x)/2
)^4*((a^3*c^4*(2205*e + 2205*f*x + 2016))/336 - (105*a^3*c^4*(e + f*x))/16) + tan(e/2 + (f*x)/2)^8*((a^3*c^4*(
3675*e + 3675*f*x + 3360))/336 - (175*a^3*c^4*(e + f*x))/16) + (7*a^3*c^4*tan(e/2 + (f*x)/2)^3)/6 + (85*a^3*c^
4*tan(e/2 + (f*x)/2)^5)/24 - (85*a^3*c^4*tan(e/2 + (f*x)/2)^9)/24 - (7*a^3*c^4*tan(e/2 + (f*x)/2)^11)/6 - (11*
a^3*c^4*tan(e/2 + (f*x)/2)^13)/8 + (a^3*c^4*(105*e + 105*f*x + 96))/336 + (11*a^3*c^4*tan(e/2 + (f*x)/2))/8 -
(5*a^3*c^4*(e + f*x))/16)/(f*(tan(e/2 + (f*x)/2)^2 + 1)^7) + (5*a^3*c^4*x)/16