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

Optimal. Leaf size=165 \[ \frac{8 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{16 a \sqrt{a \sin (e+f x)+a}}{3 f^2}-\frac{4 a x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]

[Out]

(16*a*Sqrt[a + a*Sin[e + f*x]])/(3*f^2) - (8*a*x*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (
4*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (8*a*Sin[e/2 + Pi/
4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(9*f^2)

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Rubi [A]  time = 0.0911621, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3319, 3310, 3296, 2638} \[ \frac{8 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{16 a \sqrt{a \sin (e+f x)+a}}{3 f^2}-\frac{4 a x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(16*a*Sqrt[a + a*Sin[e + f*x]])/(3*f^2) - (8*a*x*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (
4*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (8*a*Sin[e/2 + Pi/
4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(9*f^2)

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x (a+a \sin (e+f x))^{3/2} \, dx &=\left (2 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=-\frac{4 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{8 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}+\frac{1}{3} \left (4 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=-\frac{8 a x \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{4 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{8 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}+\frac{\left (8 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 f}\\ &=\frac{16 a \sqrt{a+a \sin (e+f x)}}{3 f^2}-\frac{8 a x \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{4 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{8 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}\\ \end{align*}

Mathematica [A]  time = 0.680619, size = 113, normalized size = 0.68 \[ -\frac{(a (\sin (e+f x)+1))^{3/2} \left (27 (f x-2) \cos \left (\frac{1}{2} (e+f x)\right )+(3 f x+2) \cos \left (\frac{3}{2} (e+f x)\right )+2 \sin \left (\frac{1}{2} (e+f x)\right ) ((3 f x-2) \cos (e+f x)-4 (3 f x+7))\right )}{9 f^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((27*(-2 + f*x)*Cos[(e + f*x)/2] + (2 + 3*f*x)*Cos[(3*(e + f*x))/2] + 2*(-4*(7 + 3*f*x) + (-2 + 3*f*x)*Cos[e
+ f*x])*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))/(9*f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a*(sin(e + f*x) + 1))**(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}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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