3.133 \(\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=94 \[ \frac{a B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.333473, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2971, 2738} \[ \frac{a B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2971

Int[Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[B/d, Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x
] - Dist[(B*c - A*d)/d, Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f
, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 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 \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=(A+B) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx-\frac{B \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx}{c}\\ &=-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}+\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.582407, size = 84, normalized size = 0.89 \[ \frac{c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (2 (6 A-B) \sin (e+f x)+\cos (2 (e+f x)) (3 A+2 B \sin (e+f x)-3 B))}{12 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.366, size = 91, normalized size = 1. \begin{align*}{\frac{ \left ( -2\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,A\sin \left ( fx+e \right ) -3\,B\sin \left ( fx+e \right ) -6\,A+2\,B \right ) \sin \left ( fx+e \right ) }{6\,f \left ( -1+\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) } \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{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+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2),x)

[Out]

1/6/f*(-2*B*cos(f*x+e)^2+3*A*sin(f*x+e)-3*B*sin(f*x+e)-6*A+2*B)*(-c*(-1+sin(f*x+e)))^(3/2)*sin(f*x+e)*(a*(1+si
n(f*x+e)))^(1/2)/(-1+sin(f*x+e))/cos(f*x+e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7092, size = 227, normalized size = 2.41 \begin{align*} \frac{{\left (3 \,{\left (A - B\right )} c \cos \left (f x + e\right )^{2} - 3 \,{\left (A - B\right )} c + 2 \,{\left (B c \cos \left (f x + e\right )^{2} +{\left (3 \, A - B\right )} c\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}}{6 \, f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/6*(3*(A - B)*c*cos(f*x + e)^2 - 3*(A - B)*c + 2*(B*c*cos(f*x + e)^2 + (3*A - B)*c)*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+B*sin(f*x+e))*(c-c*sin(f*x+e))**(3/2)*(a+a*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+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError