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

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

[Out]

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

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Rubi [A]  time = 0.0802235, 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} \[ \frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.266344, size = 72, normalized size = 1.67 \[ -\frac{a^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-15 \sin (e+f x)+\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^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 = 0.183, size = 80, normalized size = 1.9 \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+2\,\sin \left ( fx+e \right ) \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{5}{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))^(5/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.05643, size = 201, normalized size = 4.67 \begin{align*} -\frac{{\left (3 \, a^{2} \cos \left (f x + e\right )^{2} - 3 \, a^{2} +{\left (a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{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))^(5/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: TypeError