\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx\) [32]

Optimal result

Integrand size = 38, antiderivative size = 188 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^3 \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{35 a f \sqrt {c-c \sin (e+f x)}}+\frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{14 a f}+\frac {3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f} \]

[Out]

3/28*c*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)*(c-c*sin(f*x+e))^(3/2)/a/f+1/8*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)*(c-c
*sin(f*x+e))^(5/2)/a/f+1/35*c^3*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/f/(c-c*sin(f*x+e))^(1/2)+1/14*c^2*cos(f*x+
e)*(a+a*sin(f*x+e))^(9/2)*(c-c*sin(f*x+e))^(1/2)/a/f

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2819, 2817} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^3 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{35 a f \sqrt {c-c \sin (e+f x)}}+\frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{14 a f}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac {3 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f} \]

[In]

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(35*a*f*Sqrt[c - c*Sin[e + f*x]]) + (c^2*Cos[e + f*x]*(a + a*Sin
[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(14*a*f) + (3*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[
e + f*x])^(3/2))/(28*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(5/2))/(8*a*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))^{7/2} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac {3 \int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2} \, dx}{4 a} \\ & = \frac {3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac {(3 c) \int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2} \, dx}{7 a} \\ & = \frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{14 a f}+\frac {3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac {c^2 \int (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)} \, dx}{7 a} \\ & = \frac {c^3 \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{35 a f \sqrt {c-c \sin (e+f x)}}+\frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{14 a f}+\frac {3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.68 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {a^3 c^2 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (-1960 \cos (2 (e+f x))-980 \cos (4 (e+f x))-280 \cos (6 (e+f x))-35 \cos (8 (e+f x))+19600 \sin (e+f x)+3920 \sin (3 (e+f x))+784 \sin (5 (e+f x))+80 \sin (7 (e+f x)))}{35840 f} \]

[In]

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(a^3*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-1960*Cos[2*(e + f*x)] - 980*Cos[4*
(e + f*x)] - 280*Cos[6*(e + f*x)] - 35*Cos[8*(e + f*x)] + 19600*Sin[e + f*x] + 3920*Sin[3*(e + f*x)] + 784*Sin
[5*(e + f*x)] + 80*Sin[7*(e + f*x)]))/(35840*f)

Maple [A] (verified)

Time = 268.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (-35 \left (\cos ^{7}\left (f x +e \right )\right )+40 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+64 \cos \left (f x +e \right ) \sin \left (f x +e \right )+128 \tan \left (f x +e \right )+35 \sec \left (f x +e \right )\right )}{280 f}\)	\(110\)

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/280/f*(a*(1+sin(f*x+e)))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*a^3*c^2*(-35*cos(f*x+e)^7+40*cos(f*x+e)^5*sin(f*x+e
)+48*cos(f*x+e)^3*sin(f*x+e)+64*cos(f*x+e)*sin(f*x+e)+128*tan(f*x+e)+35*sec(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {{\left (35 \, a^{3} c^{2} \cos \left (f x + e\right )^{8} - 35 \, a^{3} c^{2} - 8 \, {\left (5 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} + 6 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} + 8 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 16 \, a^{3} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{280 \, f \cos \left (f x + e\right )} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/280*(35*a^3*c^2*cos(f*x + e)^8 - 35*a^3*c^2 - 8*(5*a^3*c^2*cos(f*x + e)^6 + 6*a^3*c^2*cos(f*x + e)^4 + 8*a^
3*c^2*cos(f*x + e)^2 + 16*a^3*c^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x
+ e))

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(5/2)*cos(f*x + e)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.34 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {32 \, {\left (35 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16} - 160 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} + 280 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 224 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} + 70 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}\right )} \sqrt {a} \sqrt {c}}{35 \, f} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

32/35*(35*a^3*c^2*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 - 160*a^3*c^2*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*p
i + 1/2*f*x + 1/2*e)^14 + 280*a^3*c^2*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 - 224*a^3*c^2*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)^10 + 70*a^3*c^2*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)^8)*sqrt(a)*sqrt(c)/f

Mupad [B] (verification not implemented)

Time = 12.96 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.00 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {{\mathrm {e}}^{-e\,8{}\mathrm {i}-f\,x\,8{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {35\,a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}-\frac {7\,a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}-\frac {7\,a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{128\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}-\frac {a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (8\,e+8\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{512\,f}+\frac {7\,a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}+\frac {7\,a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {a^3\,c^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{224\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]

[In]

int(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2),x)

[Out]

(exp(- e*8i - f*x*8i)*(c - c*sin(e + f*x))^(1/2)*((35*a^3*c^2*exp(e*8i + f*x*8i)*sin(e + f*x)*(a + a*sin(e + f
*x))^(1/2))/(32*f) - (7*a^3*c^2*exp(e*8i + f*x*8i)*cos(2*e + 2*f*x)*(a + a*sin(e + f*x))^(1/2))/(64*f) - (7*a^
3*c^2*exp(e*8i + f*x*8i)*cos(4*e + 4*f*x)*(a + a*sin(e + f*x))^(1/2))/(128*f) - (a^3*c^2*exp(e*8i + f*x*8i)*co
s(6*e + 6*f*x)*(a + a*sin(e + f*x))^(1/2))/(64*f) - (a^3*c^2*exp(e*8i + f*x*8i)*cos(8*e + 8*f*x)*(a + a*sin(e
+ f*x))^(1/2))/(512*f) + (7*a^3*c^2*exp(e*8i + f*x*8i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(32*f) + (
7*a^3*c^2*exp(e*8i + f*x*8i)*sin(5*e + 5*f*x)*(a + a*sin(e + f*x))^(1/2))/(160*f) + (a^3*c^2*exp(e*8i + f*x*8i
)*sin(7*e + 7*f*x)*(a + a*sin(e + f*x))^(1/2))/(224*f)))/(2*cos(e + f*x))