\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx\) [31]

Optimal result

Integrand size = 38, antiderivative size = 236 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f} \]

[Out]

-2/21*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/c/f-1/9*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)
*(c-c*sin(f*x+e))^(9/2)/c/f-1/9*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2)/c/f-8/315*a^4*cos(f*x
+e)*(c-c*sin(f*x+e))^(9/2)/c/f/(a+a*sin(f*x+e))^(1/2)-4/63*a^3*cos(f*x+e)*(c-c*sin(f*x+e))^(9/2)*(a+a*sin(f*x+
e))^(1/2)/c/f

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2819, 2817} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f} \]

[In]

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(-8*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt
[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(63*c*f) - (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c
- c*Sin[e + f*x])^(9/2))/(21*c*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(9*
c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*c*f)

Rule 2817

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rule 2819

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Dist[a*((2*m - 1)/(
m + n)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m
]) &&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

Rule 2920

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{9/2} \, dx}{a c} \\ & = -\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}+\frac {8 \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx}{9 c} \\ & = -\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}+\frac {(2 a) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2} \, dx}{3 c} \\ & = -\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}+\frac {\left (8 a^2\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2} \, dx}{21 c} \\ & = -\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}+\frac {\left (8 a^3\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx}{63 c} \\ & = -\frac {8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.38 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {a^3 c^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (315-420 \sin ^2(e+f x)+378 \sin ^4(e+f x)-180 \sin ^6(e+f x)+35 \sin ^8(e+f x)\right ) \tan (e+f x)}{315 f} \]

[In]

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(a^3*c^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(315 - 420*Sin[e + f*x]^2 + 378*Sin[e + f*x]^4 -
180*Sin[e + f*x]^6 + 35*Sin[e + f*x]^8)*Tan[e + f*x])/(315*f)

Maple [A] (verified)

Time = 232.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.36

method	result	size
default	\(\frac {a^{3} c^{3} \left (35 \left (\cos ^{8}\left (f x +e \right )\right )+40 \left (\cos ^{6}\left (f x +e \right )\right )+48 \left (\cos ^{4}\left (f x +e \right )\right )+64 \left (\cos ^{2}\left (f x +e \right )\right )+128\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \tan \left (f x +e \right )}{315 f}\)	\(85\)

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/315/f*a^3*c^3*(35*cos(f*x+e)^8+40*cos(f*x+e)^6+48*cos(f*x+e)^4+64*cos(f*x+e)^2+128)*(a*(1+sin(f*x+e)))^(1/2)
*(-c*(sin(f*x+e)-1))^(1/2)*tan(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.50 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {{\left (35 \, a^{3} c^{3} \cos \left (f x + e\right )^{8} + 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 48 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} c^{3}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/315*(35*a^3*c^3*cos(f*x + e)^8 + 40*a^3*c^3*cos(f*x + e)^6 + 48*a^3*c^3*cos(f*x + e)^4 + 64*a^3*c^3*cos(f*x
+ e)^2 + 128*a^3*c^3)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(7/2)*cos(f*x + e)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.07 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 \, {\left (70 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{18} - 315 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16} + 540 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 420 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 126 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}\right )} \sqrt {a} \sqrt {c}}{315 \, f} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

256/315*(70*a^3*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*
f*x + 1/2*e)^18 - 315*a^3*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4
*pi + 1/2*f*x + 1/2*e)^16 + 540*a^3*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)
)*sin(-1/4*pi + 1/2*f*x + 1/2*e)^14 - 420*a^3*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*
x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^12 + 126*a^3*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*p
i + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^10)*sqrt(a)*sqrt(c)/f

Mupad [B] (verification not implemented)

Time = 13.26 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.05 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {{\mathrm {e}}^{-e\,9{}\mathrm {i}-f\,x\,9{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {63\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}+\frac {7\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}+\frac {9\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {9\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{896\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (9\,e+9\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{1152\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]

[In]

int(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(7/2),x)

[Out]

(exp(- e*9i - f*x*9i)*(c - c*sin(e + f*x))^(1/2)*((63*a^3*c^3*exp(e*9i + f*x*9i)*sin(e + f*x)*(a + a*sin(e + f
*x))^(1/2))/(64*f) + (7*a^3*c^3*exp(e*9i + f*x*9i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(32*f) + (9*a^
3*c^3*exp(e*9i + f*x*9i)*sin(5*e + 5*f*x)*(a + a*sin(e + f*x))^(1/2))/(160*f) + (9*a^3*c^3*exp(e*9i + f*x*9i)*
sin(7*e + 7*f*x)*(a + a*sin(e + f*x))^(1/2))/(896*f) + (a^3*c^3*exp(e*9i + f*x*9i)*sin(9*e + 9*f*x)*(a + a*sin
(e + f*x))^(1/2))/(1152*f)))/(2*cos(e + f*x))