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

Optimal. Leaf size=221 \[ \frac{3}{2} a \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{1}{2} a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{1}{2} a \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{2} a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a} \]

[Out]

(a*Cos[(3*(2*e - Pi))/4]*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*C
osIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*Cos[(2*e +
Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/2 - (a*Csc[e/2 + Pi/4 + (f*x)/
2]*Sin[(3*(2*e - Pi))/4]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2])/2

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Rubi [A]  time = 0.275993, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3312, 3303, 3299, 3302} \[ \frac{3}{2} a \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{1}{2} a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{1}{2} a \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{2} a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)/x,x]

[Out]

(a*Cos[(3*(2*e - Pi))/4]*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*C
osIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/2 + (3*a*Cos[(2*e +
Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/2 - (a*Csc[e/2 + Pi/4 + (f*x)/
2]*Sin[(3*(2*e - Pi))/4]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2])/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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3303

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.662187, size = 127, normalized size = 0.57 \[ \frac{(a (\sin (e+f x)+1))^{3/2} \left (3 \text{CosIntegral}\left (\frac{f x}{2}\right ) \left (\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )\right )+\text{CosIntegral}\left (\frac{3 f x}{2}\right ) \left (\sin \left (\frac{3 e}{2}\right )-\cos \left (\frac{3 e}{2}\right )\right )+\left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left ((2 \sin (e)+1) \text{Si}\left (\frac{3 f x}{2}\right )+3 \text{Si}\left (\frac{f x}{2}\right )\right )\right )}{2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(3/2)*(3*CosIntegral[(f*x)/2]*(Cos[e/2] + Sin[e/2]) + CosIntegral[(3*f*x)/2]*(-Cos[(3*
e)/2] + Sin[(3*e)/2]) + (Cos[e/2] - Sin[e/2])*(3*SinIntegral[(f*x)/2] + (1 + 2*Sin[e])*SinIntegral[(3*f*x)/2])
))/(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{\frac{1}{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((a+a*sin(f*x+e))^(3/2)/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+a*sin(f*x+e))^(3/2)/x,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((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="giac")

[Out]

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