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

[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)

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Rubi [A]  time = 0.734853, antiderivative size = 236, normalized size of antiderivative = 1., 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 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]

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 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)]

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 = 1.31127, 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)

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Maple [A]  time = 0.231, size = 87, normalized size = 0.4 \begin{align*}{\frac{ \left ( 35\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+40\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+48\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+64\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+128 \right ) \sin \left ( fx+e \right ) }{315\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)

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Fricas [A]  time = 1.93308, size = 289, normalized size = 1.22 \begin{align*} \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 )} \end{align*}

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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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]

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