3.532 \(\int (a+a \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]

[Out]

(-8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)

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Rubi [A]  time = 0.0294619, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(-8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^{3/2} \, dx &=-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{1}{3} (4 a) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.142243, size = 89, normalized size = 1.51 \[ -\frac{(a (\sin (e+f x)+1))^{3/2} \left (-9 \sin \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{3}{2} (e+f x)\right )+9 \cos \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{3}{2} (e+f x)\right )\right )}{3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(3/2),x]

[Out]

-((a*(1 + Sin[e + f*x]))^(3/2)*(9*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] - 9*Sin[(e + f*x)/2] + Sin[(3*(e + f
*x))/2]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

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Maple [A]  time = 0.46, size = 53, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{2} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( \sin \left ( fx+e \right ) +5 \right ) }{3\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(3/2),x)

[Out]

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

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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{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2), x)

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Fricas [A]  time = 1.51968, size = 204, normalized size = 3.46 \begin{align*} -\frac{2 \,{\left (a \cos \left (f x + e\right )^{2} + 5 \, a \cos \left (f x + e\right ) +{\left (a \cos \left (f x + e\right ) - 4 \, a\right )} \sin \left (f x + e\right ) + 4 \, a\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{3 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (e + f x \right )} + a\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral((a*sin(e + f*x) + a)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2), x)