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

Optimal. Leaf size=43 \[ \frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}} \]

[Out]

(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 0.0815774, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2738} \[ \frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])

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} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.340935, size = 82, normalized size = 1.91 \[ \frac{a^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (56 \sin (e+f x)-8 \sin (3 (e+f x))-28 \cos (2 (e+f x))+\cos (4 (e+f x)))}{32 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-28*Cos[2*(e + f*x)] + Cos[4*(e + f*x)]
+ 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f)

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Maple [B]  time = 0.17, size = 103, normalized size = 2.4 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( - \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}+4\,\sin \left ( fx+e \right ) -4 \right ) }{4\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \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))^(1/2),x)

[Out]

-1/4/f*(-c*(-1+sin(f*x+e)))^(1/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)*(-cos(f*x+e)^6+sin(f*x+e)*cos(f*x+e)^4+c
os(f*x+e)^2*sin(f*x+e)+cos(f*x+e)^2+4*sin(f*x+e)-4)/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}} \sqrt{-c \sin \left (f x + e\right ) + c}\,{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))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.1097, size = 231, normalized size = 5.37 \begin{align*} \frac{{\left (a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 7 \, a^{3} - 4 \,{\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{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}}{4 \, 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))^(1/2),x, algorithm="fricas")

[Out]

1/4*(a^3*cos(f*x + e)^4 - 8*a^3*cos(f*x + e)^2 + 7*a^3 - 4*(a^3*cos(f*x + e)^2 - 2*a^3)*sin(f*x + e))*sqrt(a*s
in(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))**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError