3.271 \(\int (\sin (x \log (x))+\log (x) \sin (x \log (x))) \, dx\)

Optimal. Leaf size=7 \[ -\cos (x \log (x)) \]

[Out]

-Cos[x*Log[x]]

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Rubi [A]  time = 0.0269251, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4511} \[ -\cos (x \log (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x*Log[x]] + Log[x]*Sin[x*Log[x]],x]

[Out]

-Cos[x*Log[x]]

Rule 4511

Int[Log[(b_.)*(x_)]*Sin[Log[(b_.)*(x_)]*(a_.)*(x_)], x_Symbol] :> -Simp[Cos[a*x*Log[b*x]]/a, x] - Int[Sin[a*x*
Log[b*x]], x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int (\sin (x \log (x))+\log (x) \sin (x \log (x))) \, dx &=\int \sin (x \log (x)) \, dx+\int \log (x) \sin (x \log (x)) \, dx\\ &=-\cos (x \log (x))\\ \end{align*}

Mathematica [A]  time = 0.108954, size = 7, normalized size = 1. \[ -\cos (x \log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x*Log[x]] + Log[x]*Sin[x*Log[x]],x]

[Out]

-Cos[x*Log[x]]

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Maple [A]  time = 0.017, size = 8, normalized size = 1.1 \begin{align*} -\cos \left ( x\ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x*ln(x))+ln(x)*sin(x*ln(x)),x)

[Out]

-cos(x*ln(x))

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Maxima [A]  time = 1.2649, size = 9, normalized size = 1.29 \begin{align*} -\cos \left (x \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x*log(x))+log(x)*sin(x*log(x)),x, algorithm="maxima")

[Out]

-cos(x*log(x))

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Fricas [A]  time = 2.19406, size = 22, normalized size = 3.14 \begin{align*} -\cos \left (x \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x*log(x))+log(x)*sin(x*log(x)),x, algorithm="fricas")

[Out]

-cos(x*log(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\log{\left (x \right )} + 1\right ) \sin{\left (x \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x*ln(x))+ln(x)*sin(x*ln(x)),x)

[Out]

Integral((log(x) + 1)*sin(x*log(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x*log(x))+log(x)*sin(x*log(x)),x, algorithm="giac")

[Out]

integrate(log(x)*sin(x*log(x)) + sin(x*log(x)), x)