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*ln(x))

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Rubi [A]  time = 0.03, antiderivative size = 7, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.11, size = 7, normalized size = 1.00 \[ -\cos (x \log (x)) \]

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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fricas [A]  time = 0.43, size = 7, normalized size = 1.00 \[ -\cos \left (x \log \relax (x)\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
9.46Not invertible Error: Bad Argument Value

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maple [A]  time = 0.08, size = 8, normalized size = 1.14 \[ -\cos \left (x \ln \relax (x )\right ) \]

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 = 1.09, size = 7, normalized size = 1.00 \[ -\cos \left (x \log \relax (x)\right ) \]

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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mupad [B]  time = 0.42, size = 7, normalized size = 1.00 \[ -\cos \left (x\,\ln \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x*log(x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\log {\relax (x )} + 1\right ) \sin {\left (x \log {\relax (x )} \right )}\, dx \]

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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