### 3.51 $$\int \sin (\log (x)) \, dx$$

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0030924, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {4475} $\frac{1}{2} x \sin (\log (x))-\frac{1}{2} x \cos (\log (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 4475

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*}

Mathematica [A]  time = 0.0019087, size = 17, normalized size = 1. $\frac{1}{2} x \sin (\log (x))-\frac{1}{2} x \cos (\log (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*} -{\frac{x\cos \left ( \ln \left ( x \right ) \right ) }{2}}+{\frac{x\sin \left ( \ln \left ( x \right ) \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(ln(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.93754, size = 16, normalized size = 0.94 \begin{align*} -\frac{1}{2} \, x{\left (\cos \left (\log \left (x\right )\right ) - \sin \left (\log \left (x\right )\right )\right )} \end{align*}

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)))

________________________________________________________________________________________

Fricas [A]  time = 2.35962, size = 54, normalized size = 3.18 \begin{align*} -\frac{1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac{1}{2} \, x \sin \left (\log \left (x\right )\right ) \end{align*}

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))

________________________________________________________________________________________

Sympy [A]  time = 0.374718, size = 15, normalized size = 0.88 \begin{align*} \frac{x \sin{\left (\log{\left (x \right )} \right )}}{2} - \frac{x \cos{\left (\log{\left (x \right )} \right )}}{2} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.04808, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac{1}{2} \, x \sin \left (\log \left (x\right )\right ) \end{align*}

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))