3.1.51 \(\int \sin (\log (x)) \, dx\) [51]

Optimal. Leaf size=17 \[ -\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \]

[Out]

-1/2*x*cos(ln(x))+1/2*x*sin(ln(x))

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4563} \begin {gather*} \frac {1}{2} x \sin (\log (x))-\frac {1}{2} x \cos (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[Log[x]],x]

[Out]

-1/2*(x*Cos[Log[x]]) + (x*Sin[Log[x]])/2

Rule 4563

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*(Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*
n^2 + 1)), x] - Simp[b*d*n*x*(Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1)), x] /; FreeQ[{a, b, c, d, n}, x] &&
NeQ[b^2*d^2*n^2 + 1, 0]

Rubi steps

\begin {align*} \int \sin (\log (x)) \, dx &=-\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x))\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[Log[x]],x]

[Out]

-1/2*(x*Cos[Log[x]]) + (x*Sin[Log[x]])/2

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
lookup	\(-\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
default	\(-\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
risch	\(\left (-\frac {1}{4}-\frac {i}{4}\right ) x \,x^{i}+\left (-\frac {1}{4}+\frac {i}{4}\right ) x \,x^{-i}\)	\(22\)
norman	\(\frac {x \tan \left (\frac {\ln \left (x \right )}{2}\right )-\frac {x}{2}+\frac {x \left (\tan ^{2}\left (\frac {\ln \left (x \right )}{2}\right )\right )}{2}}{1+\tan ^{2}\left (\frac {\ln \left (x \right )}{2}\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(ln(x)),x,method=_RETURNVERBOSE)

[Out]

-1/2*x*cos(ln(x))+1/2*x*sin(ln(x))

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Maxima [A]
time = 4.46, size = 12, normalized size = 0.71 \begin {gather*} -\frac {1}{2} \, x {\left (\cos \left (\log \left (x\right )\right ) - \sin \left (\log \left (x\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*(cos(log(x)) - sin(log(x)))

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Fricas [A]
time = 0.88, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*cos(log(x)) + 1/2*x*sin(log(x))

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Sympy [A]
time = 0.11, size = 15, normalized size = 0.88 \begin {gather*} \frac {x \sin {\left (\log {\left (x \right )} \right )}}{2} - \frac {x \cos {\left (\log {\left (x \right )} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(ln(x)),x)

[Out]

x*sin(log(x))/2 - x*cos(log(x))/2

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Giac [A]
time = 0.47, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*cos(log(x)) + 1/2*x*sin(log(x))

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {2}\,x\,\cos \left (\frac {\pi }{4}+\ln \left (x\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(log(x)),x)

[Out]

-(2^(1/2)*x*cos(pi/4 + log(x)))/2

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