\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx\) [101]

Optimal result

Integrand size = 38, antiderivative size = 81 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^3 (9 A+5 B) c^4 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \]

[Out]

2/63*a^3*(9*A+5*B)*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)-2/9*a^3*B*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(5/
2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {3046, 2935, 2752} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^3 c^4 (9 A+5 B) \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(2*a^3*(9*A + 5*B)*c^4*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(9*f*(
c - c*Sin[e + f*x])^(5/2))

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2935

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p +
1, 0]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[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 \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = -\frac {2 a^3 B c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}+\frac {1}{9} \left (a^3 (9 A+5 B) c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {2 a^3 (9 A+5 B) c^4 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (9 A-2 B+7 B \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{63 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

(2*a^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(9*A - 2*B + 7*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(63*f*
(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (7 B \sin \left (f x +e \right )+9 A -2 B \right )}{63 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(65\)
parts	\(-\frac {2 A \,a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right ) c}{\cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 B \,a^{3} \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )-40 \left (\sin ^{3}\left (f x +e \right )\right )+48 \left (\sin ^{2}\left (f x +e \right )\right )-64 \sin \left (f x +e \right )+128\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (A +3 B \right ) \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-6 \left (\sin ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-16\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (3 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-2\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )+8\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(345\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/63*(sin(f*x+e)-1)*c*(1+sin(f*x+e))^4*a^3*(7*B*sin(f*x+e)+9*A-2*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (73) = 146\).

Time = 0.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.86 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, {\left (7 \, B a^{3} \cos \left (f x + e\right )^{5} + {\left (9 \, A + 26 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - {\left (27 \, A + 29 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 4 \, {\left (18 \, A + 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (9 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right ) + 8 \, {\left (9 \, A + 5 \, B\right )} a^{3} + {\left (7 \, B a^{3} \cos \left (f x + e\right )^{4} - {\left (9 \, A + 19 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 12 \, {\left (3 \, A + 4 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (9 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right ) + 8 \, {\left (9 \, A + 5 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{63 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/63*(7*B*a^3*cos(f*x + e)^5 + (9*A + 26*B)*a^3*cos(f*x + e)^4 - (27*A + 29*B)*a^3*cos(f*x + e)^3 - 4*(18*A +
17*B)*a^3*cos(f*x + e)^2 + 4*(9*A + 5*B)*a^3*cos(f*x + e) + 8*(9*A + 5*B)*a^3 + (7*B*a^3*cos(f*x + e)^4 - (9*A
+ 19*B)*a^3*cos(f*x + e)^3 - 12*(3*A + 4*B)*a^3*cos(f*x + e)^2 + 4*(9*A + 5*B)*a^3*cos(f*x + e) + 8*(9*A + 5*
B)*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

Sympy [F]

\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=a^{3} \left (\int A \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int 3 A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int 3 A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int 3 B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(1/2),x)

[Out]

a**3*(Integral(A*sqrt(-c*sin(e + f*x) + c), x) + Integral(3*A*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + Int
egral(3*A*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2, x) + Integral(A*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3
, x) + Integral(B*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + Integral(3*B*sqrt(-c*sin(e + f*x) + c)*sin(e +
f*x)**2, x) + Integral(3*B*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3, x) + Integral(B*sqrt(-c*sin(e + f*x) + c
)*sin(e + f*x)**4, x))

Maxima [F]

\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*sqrt(-c*sin(f*x + e) + c), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (73) = 146\).

Time = 0.44 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.12 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (7 \, B a^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 126 \, {\left (5 \, A a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, B a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 42 \, {\left (9 \, A a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 126 \, {\left (A a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 9 \, {\left (2 \, A a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right )\right )} \sqrt {c}}{504 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-1/504*sqrt(2)*(7*B*a^3*cos(-9/4*pi + 9/2*f*x + 9/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 126*(5*A*a^3*sgn(
sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*B*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)
+ 42*(9*A*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi +
3/2*f*x + 3/2*e) + 126*(A*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))
)*cos(-5/4*pi + 5/2*f*x + 5/2*e) + 9*(2*A*a^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^3*sgn(sin(-1/4*pi +
1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e))*sqrt(c)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2), x)