3.105 \(\int \frac{\sin ^2(x)}{x} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]

[Out]

-CosIntegral[2*x]/2 + Log[x]/2

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Rubi [A]  time = 0.0554678, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^2/x,x]

[Out]

-CosIntegral[2*x]/2 + Log[x]/2

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Rubi in Sympy [A]  time = 2.13974, size = 10, normalized size = 0.67 \[ \frac{\log{\left (x \right )}}{2} - \frac{\operatorname{Ci}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**2/x,x)

[Out]

log(x)/2 - Ci(2*x)/2

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Mathematica [A]  time = 0.00742617, size = 15, normalized size = 1. \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^2/x,x]

[Out]

-CosIntegral[2*x]/2 + Log[x]/2

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Maple [A]  time = 0.008, size = 12, normalized size = 0.8 \[ -{\frac{{\it Ci} \left ( 2\,x \right ) }{2}}+{\frac{\ln \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^2/x,x)

[Out]

-1/2*Ci(2*x)+1/2*ln(x)

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Maxima [A]  time = 1.47929, size = 23, normalized size = 1.53 \[ -\frac{1}{4} \,{\rm Ei}\left (2 i \, x\right ) - \frac{1}{4} \,{\rm Ei}\left (-2 i \, x\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/x,x, algorithm="maxima")

[Out]

-1/4*Ei(2*I*x) - 1/4*Ei(-2*I*x) + 1/2*log(x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{4} \, \operatorname{Ci}\left (2 \, x\right ) - \frac{1}{4} \, \operatorname{Ci}\left (-2 \, x\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/x,x, algorithm="fricas")

[Out]

-1/4*cos_integral(2*x) - 1/4*cos_integral(-2*x) + 1/2*log(x)

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Sympy [A]  time = 2.23092, size = 10, normalized size = 0.67 \[ \frac{\log{\left (x \right )}}{2} - \frac{\operatorname{Ci}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**2/x,x)

[Out]

log(x)/2 - Ci(2*x)/2

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GIAC/XCAS [A]  time = 0.199283, size = 15, normalized size = 1. \[ -\frac{1}{2} \,{\rm Ci}\left (2 \, x\right ) + \frac{1}{2} \,{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/x,x, algorithm="giac")

[Out]

-1/2*Ci(2*x) + 1/2*ln(x)