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

Optimal. Leaf size=134 \[ \frac{2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f} \]

[Out]

(c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(15*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(a + a*Sin
[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e +
f*x])^(3/2))/(6*f)

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Rubi [A]  time = 0.263819, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ \frac{2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f} \]

Antiderivative was successfully verified.

[In]

Int[(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])^(7/2))/(15*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(a + a*Sin
[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e +
f*x])^(3/2))/(6*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))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx &=\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}+\frac{1}{3} (2 c) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}+\frac{1}{15} \left (4 c^2\right ) \int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{c^3 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}\\ \end{align*}

Mathematica [A]  time = 1.22532, size = 107, normalized size = 0.8 \[ \frac{a^3 c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]]*(-75*Cos[2*(e + f*x)] - 30*Cos[4*(e
+ f*x)] - 5*Cos[6*(e + f*x)] + 600*Sin[e + f*x] + 100*Sin[3*(e + f*x)] + 12*Sin[5*(e + f*x)]))/(960*f)

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Maple [A]  time = 0.167, size = 116, normalized size = 0.9 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+11\,\sin \left ( fx+e \right ) -11 \right ) }{30\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{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((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x)

[Out]

-1/30/f*(-c*(-1+sin(f*x+e)))^(5/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)*(-5*cos(f*x+e)^6+sin(f*x+e)*cos(f*x+e)^
4-6*cos(f*x+e)^4+3*cos(f*x+e)^2*sin(f*x+e)-8*cos(f*x+e)^2+11*sin(f*x+e)-11)/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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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), x)

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Fricas [A]  time = 1.1617, size = 266, normalized size = 1.99 \begin{align*} -\frac{{\left (5 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} - 5 \, a^{3} c^{2} - 2 \,{\left (3 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 8 \, 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}}{30 \, 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))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/30*(5*a^3*c^2*cos(f*x + e)^6 - 5*a^3*c^2 - 2*(3*a^3*c^2*cos(f*x + e)^4 + 4*a^3*c^2*cos(f*x + e)^2 + 8*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))

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2