3.348 \(\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\)

Optimal. Leaf size=89 \[ -\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f} \]

[Out]

-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Si
n[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*f)

________________________________________________________________________________________

Rubi [A]  time = 0.187908, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ -\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Si
n[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*f)

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 (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx &=-\frac{a \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{5 f}+\frac{1}{5} (2 a) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{5 f}\\ \end{align*}

Mathematica [A]  time = 1.02234, size = 146, normalized size = 1.64 \[ -\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (70 \sin (e+f x)+5 \sin (3 (e+f x))-\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

-(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos
[4*(e + f*x)] + 70*Sin[e + f*x] + 5*Sin[3*(e + f*x)] - Sin[5*(e + f*x)]))/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f
*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

________________________________________________________________________________________

Maple [A]  time = 0.168, size = 106, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( 2\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+2\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +6\,\sin \left ( fx+e \right ) +6 \right ) }{10\,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{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2),x)

[Out]

1/10/f*(-c*(-1+sin(f*x+e)))^(7/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(3/2)*(2*cos(f*x+e)^6+sin(f*x+e)*cos(f*x+e)^4+
2*cos(f*x+e)^4+3*cos(f*x+e)^2*sin(f*x+e)+6*sin(f*x+e)+6)/cos(f*x+e)^7

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*(-c*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.46939, size = 248, normalized size = 2.79 \begin{align*} \frac{{\left (5 \, a c^{3} \cos \left (f x + e\right )^{4} - 5 \, a c^{3} - 2 \,{\left (a c^{3} \cos \left (f x + e\right )^{4} - 2 \, a c^{3} \cos \left (f x + e\right )^{2} - 4 \, a c^{3}\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}}{10 \, f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/10*(5*a*c^3*cos(f*x + e)^4 - 5*a*c^3 - 2*(a*c^3*cos(f*x + e)^4 - 2*a*c^3*cos(f*x + e)^2 - 4*a*c^3)*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)]  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((a+a*sin(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*(-c*sin(f*x + e) + c)^(7/2), x)