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

Optimal. Leaf size=43 \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}} \]

[Out]

-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.0820688, 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} \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*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 \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.286207, size = 74, normalized size = 1.72 \[ \frac{c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (15 \sin (e+f x)-\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] -
Sin[3*(e + f*x)]))/(12*f)

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Maple [B]  time = 0.179, size = 78, normalized size = 1.8 \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +2\,\sin \left ( fx+e \right ) +2 \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x)

[Out]

1/3/f*(-c*(-1+sin(f*x+e)))^(5/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(1/2)*(cos(f*x+e)^4+cos(f*x+e)^2*sin(f*x+e)+2*s
in(f*x+e)+2)/cos(f*x+e)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (f x + e\right ) + a}{\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))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(5/2), x)

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Fricas [B]  time = 1.10867, size = 200, normalized size = 4.65 \begin{align*} \frac{{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} -{\left (c^{2} \cos \left (f x + e\right )^{2} - 4 \, 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}}{3 \, 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))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/3*(3*c^2*cos(f*x + e)^2 - 3*c^2 - (c^2*cos(f*x + e)^2 - 4*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))**(1/2)*(c-c*sin(f*x+e))**(5/2),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))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError