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

Optimal. Leaf size=236 \[ -\frac {8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f} \]

[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]  time = 0.73, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ -\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac {8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt {a \sin (e+f x)+a}}-\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f} \]

Antiderivative was successfully verified.

[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 2738

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 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(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] && E
qQ[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 2841

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*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx &=\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]  time = 1.26, size = 97, normalized size = 0.41 \[ \frac {a^3 c^3 (39690 \sin (e+f x)+8820 \sin (3 (e+f x))+2268 \sin (5 (e+f x))+405 \sin (7 (e+f x))+35 \sin (9 (e+f x))) \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)}}{80640 f} \]

Antiderivative was successfully verified.

[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*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(39690*Sin[e + f*x] + 8820*Sin[3*(e
+ f*x)] + 2268*Sin[5*(e + f*x)] + 405*Sin[7*(e + f*x)] + 35*Sin[9*(e + f*x)]))/(80640*f)

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 117, normalized size = 0.50 \[ \frac {{\left (35 \, a^{3} c^{3} \cos \left (f x + e\right )^{8} + 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 48 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} c^{3}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*sqrt(2*c)*(-16128*a^3*c^3*f*si
gn(sin(1/2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(f*x+exp(1))/(256*f)^2-8064*a^3*c^3*f*s
ign(sin(1/2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(3*f*x+3*exp(1))/(384*f)^2-5760*a^3*c^
3*f*sign(sin(1/2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(5*f*x+5*exp(1))/(640*f)^2-32256*
a^3*c^3*f*sign(sin(1/2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(7*f*x+7*exp(1))/(3584*f)^2
-4608*a^3*c^3*f*sign(sin(1/2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(9*f*x+9*exp(1))/(460
8*f)^2)

________________________________________________________________________________________

maple [A]  time = 0.39, size = 87, normalized size = 0.37 \[ \frac {\left (35 \left (\cos ^{8}\left (f x +e \right )\right )+40 \left (\cos ^{6}\left (f x +e \right )\right )+48 \left (\cos ^{4}\left (f x +e \right )\right )+64 \left (\cos ^{2}\left (f x +e \right )\right )+128\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{315 f \cos \left (f x +e \right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315/f*(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)*(-c*(sin(f*x+e)-1))^(7/2)*sin(f*
x+e)*(a*(1+sin(f*x+e)))^(7/2)/cos(f*x+e)^7

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 12.12, size = 247, normalized size = 1.05 \[ \frac {{\mathrm {e}}^{-e\,9{}\mathrm {i}-f\,x\,9{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {63\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}+\frac {7\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}+\frac {9\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {9\,a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{896\,f}+\frac {a^3\,c^3\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sin \left (9\,e+9\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{1152\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________