3.114 \(\int \frac{\sin ^2(a+\frac{b}{x})}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} \cos (2 a) \text{CosIntegral}\left (\frac{2 b}{x}\right )-\frac{1}{2} \sin (2 a) \text{Si}\left (\frac{2 b}{x}\right )+\frac{\log (x)}{2} \]

[Out]

(Cos[2*a]*CosIntegral[(2*b)/x])/2 + Log[x]/2 - (Sin[2*a]*SinIntegral[(2*b)/x])/2

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Rubi [A]  time = 0.049541, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3425, 3378, 3376, 3375} \[ \frac{1}{2} \cos (2 a) \text{CosIntegral}\left (\frac{2 b}{x}\right )-\frac{1}{2} \sin (2 a) \text{Si}\left (\frac{2 b}{x}\right )+\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x]^2/x,x]

[Out]

(Cos[2*a]*CosIntegral[(2*b)/x])/2 + Log[x]/2 - (Sin[2*a]*SinIntegral[(2*b)/x])/2

Rule 3425

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rule 3378

Int[Cos[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cos[c], Int[Cos[d*x^n]/x, x], x] - Dist[Sin[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

\begin{align*} \int \frac{\sin ^2\left (a+\frac{b}{x}\right )}{x} \, dx &=\int \left (\frac{1}{2 x}-\frac{\cos \left (2 a+\frac{2 b}{x}\right )}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}-\frac{1}{2} \int \frac{\cos \left (2 a+\frac{2 b}{x}\right )}{x} \, dx\\ &=\frac{\log (x)}{2}-\frac{1}{2} \cos (2 a) \int \frac{\cos \left (\frac{2 b}{x}\right )}{x} \, dx+\frac{1}{2} \sin (2 a) \int \frac{\sin \left (\frac{2 b}{x}\right )}{x} \, dx\\ &=\frac{1}{2} \cos (2 a) \text{Ci}\left (\frac{2 b}{x}\right )+\frac{\log (x)}{2}-\frac{1}{2} \sin (2 a) \text{Si}\left (\frac{2 b}{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0586643, size = 32, normalized size = 0.86 \[ \frac{1}{2} \left (\cos (2 a) \text{CosIntegral}\left (\frac{2 b}{x}\right )-\sin (2 a) \text{Si}\left (\frac{2 b}{x}\right )+\log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x]^2/x,x]

[Out]

(Cos[2*a]*CosIntegral[(2*b)/x] + Log[x] - Sin[2*a]*SinIntegral[(2*b)/x])/2

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Maple [A]  time = 0.011, size = 36, normalized size = 1. \begin{align*} -{\frac{1}{2}\ln \left ({\frac{b}{x}} \right ) }-{\frac{\sin \left ( 2\,a \right ) }{2}{\it Si} \left ( 2\,{\frac{b}{x}} \right ) }+{\frac{\cos \left ( 2\,a \right ) }{2}{\it Ci} \left ( 2\,{\frac{b}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/x)^2/x,x)

[Out]

-1/2*ln(b/x)-1/2*Si(2*b/x)*sin(2*a)+1/2*Ci(2*b/x)*cos(2*a)

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Maxima [C]  time = 1.12719, size = 69, normalized size = 1.86 \begin{align*} \frac{1}{4} \,{\left ({\rm Ei}\left (\frac{2 i \, b}{x}\right ) +{\rm Ei}\left (-\frac{2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) + \frac{1}{4} \,{\left (i \,{\rm Ei}\left (\frac{2 i \, b}{x}\right ) - i \,{\rm Ei}\left (-\frac{2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)^2/x,x, algorithm="maxima")

[Out]

1/4*(Ei(2*I*b/x) + Ei(-2*I*b/x))*cos(2*a) + 1/4*(I*Ei(2*I*b/x) - I*Ei(-2*I*b/x))*sin(2*a) + 1/2*log(x)

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Fricas [A]  time = 1.43462, size = 144, normalized size = 3.89 \begin{align*} \frac{1}{4} \,{\left (\operatorname{Ci}\left (\frac{2 \, b}{x}\right ) + \operatorname{Ci}\left (-\frac{2 \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - \frac{1}{2} \, \sin \left (2 \, a\right ) \operatorname{Si}\left (\frac{2 \, b}{x}\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)^2/x,x, algorithm="fricas")

[Out]

1/4*(cos_integral(2*b/x) + cos_integral(-2*b/x))*cos(2*a) - 1/2*sin(2*a)*sin_integral(2*b/x) + 1/2*log(x)

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Sympy [A]  time = 4.6788, size = 31, normalized size = 0.84 \begin{align*} \frac{\log{\left (x \right )}}{2} - \frac{\sin{\left (2 a \right )} \operatorname{Si}{\left (\frac{2 b}{x} \right )}}{2} + \frac{\cos{\left (2 a \right )} \operatorname{Ci}{\left (\frac{2 b}{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)**2/x,x)

[Out]

log(x)/2 - sin(2*a)*Si(2*b/x)/2 + cos(2*a)*Ci(2*b/x)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)^2/x,x, algorithm="giac")

[Out]

integrate(sin(a + b/x)^2/x, x)