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

Optimal. Leaf size=263 \[ -\frac{3}{4} a f \sin \left (\frac{1}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \sin \left (\frac{1}{4} (6 e+\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{3}{4} a f \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \cos \left (\frac{1}{4} (6 e+\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{x} \]

[Out]

(-3*a*f*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e - Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/4 + (3*a*f*
CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(6*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/4 - (2*a*Sin[e/2
+ Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/x - (3*a*f*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a +
a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/4 + (3*a*f*Cos[(6*e + Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e
+ f*x]]*SinIntegral[(3*f*x)/2])/4

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Rubi [A]  time = 0.300412, antiderivative size = 263, 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, 3313, 3303, 3299, 3302} \[ -\frac{3}{4} a f \sin \left (\frac{1}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \sin \left (\frac{1}{4} (6 e+\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{3}{4} a f \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \cos \left (\frac{1}{4} (6 e+\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)/x^2,x]

[Out]

(-3*a*f*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e - Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/4 + (3*a*f*
CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(6*e + Pi)/4]*Sqrt[a + a*Sin[e + f*x]])/4 - (2*a*Sin[e/2
+ Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/x - (3*a*f*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(2*e + Pi)/4]*Sqrt[a +
a*Sin[e + f*x]]*SinIntegral[(f*x)/2])/4 + (3*a*f*Cos[(6*e + Pi)/4]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e
+ f*x]]*SinIntegral[(3*f*x)/2])/4

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 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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^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 \frac{\sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{x^2} \, dx\\ &=-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x}+\left (3 a f \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \left (\frac{\cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{4 x}+\frac{\cos \left (\frac{3 e}{2}-\frac{\pi }{4}+\frac{3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x}+\frac{1}{4} \left (3 a f \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{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{x} \, dx+\frac{1}{4} \left (3 a f \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 e}{2}-\frac{\pi }{4}+\frac{3 f x}{2}\right )}{x} \, dx\\ &=-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x}+\frac{1}{4} \left (3 a f \cos \left (\frac{1}{4} (6 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{3 f x}{2}\right )}{x} \, dx-\frac{1}{4} \left (3 a f \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}{4} \left (3 a f \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{\sin \left (\frac{f x}{2}\right )}{x} \, dx+\frac{1}{4} \left (3 a f \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{1}{4} (6 e+\pi )\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\cos \left (\frac{3 f x}{2}\right )}{x} \, dx\\ &=-\frac{3}{4} a f \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}{4} a f \text{Ci}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{1}{4} (6 e+\pi )\right ) \sqrt{a+a \sin (e+f x)}-\frac{2 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x}-\frac{3}{4} a f \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)} \text{Si}\left (\frac{f x}{2}\right )+\frac{3}{4} a f \cos \left (\frac{1}{4} (6 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{3 f x}{2}\right )\\ \end{align*}

Mathematica [C]  time = 0.916471, size = 226, normalized size = 0.86 \[ \frac{i \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2} \left (3 f x e^{i e+\frac{3 i f x}{2}} \text{Ei}\left (-\frac{1}{2} i f x\right )+3 i f x e^{2 i e+\frac{3 i f x}{2}} \text{Ei}\left (\frac{i f x}{2}\right )+3 f x e^{\frac{3}{2} i (2 e+f x)} \text{Ei}\left (\frac{3 i f x}{2}\right )-6 i e^{i (e+f x)}-6 e^{2 i (e+f x)}+2 i e^{3 i (e+f x)}+3 i f x e^{\frac{3 i f x}{2}} \text{Ei}\left (-\frac{3}{2} i f x\right )+2\right )}{4 \sqrt{2} x \left (e^{i (e+f x)}+i\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(2 - (6*I)*E^(I*(e + f*x)) - 6*E^((2*I)*(e + f
*x)) + (2*I)*E^((3*I)*(e + f*x)) + 3*E^(I*e + ((3*I)/2)*f*x)*f*x*ExpIntegralEi[(-I/2)*f*x] + (3*I)*E^((2*I)*e
+ ((3*I)/2)*f*x)*f*x*ExpIntegralEi[(I/2)*f*x] + (3*I)*E^(((3*I)/2)*f*x)*f*x*ExpIntegralEi[((-3*I)/2)*f*x] + 3*
E^(((3*I)/2)*(2*e + f*x))*f*x*ExpIntegralEi[((3*I)/2)*f*x]))/(Sqrt[2]*(I + E^(I*(e + f*x)))^3*x)

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

[Out]

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

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)/x**2, 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^{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^2,x, algorithm="giac")

[Out]

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