3.1.77 \(\int \frac {\cos (x) \log (x)-\frac {\sin (x)}{x}}{\log ^2(x)} \, dx\) [77]

Optimal. Leaf size=7 \[ \frac {\sin (x)}{\log (x)} \]

[Out]

sin(x)/ln(x)

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Rubi [F]
time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cos (x) \log (x)-\frac {\sin (x)}{x}}{\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Cos[x]*Log[x] - Sin[x]/x)/Log[x]^2,x]

[Out]

Defer[Int][Cos[x]/Log[x], x] - Defer[Int][Sin[x]/(x*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (\frac {\cos (x)}{\log (x)}-\frac {\sin (x)}{x \log ^2(x)}\right ) \, dx\\ &=\int \frac {\cos (x)}{\log (x)} \, dx-\int \frac {\sin (x)}{x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.45, size = 7, normalized size = 1.00 \begin {gather*} \frac {\sin (x)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]*Log[x] - Sin[x]/x)/Log[x]^2,x]

[Out]

Sin[x]/Log[x]

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Maple [A]
time = 0.12, size = 8, normalized size = 1.14

method	result	size
risch	\(\frac {\sin \left (x \right )}{\ln \left (x \right )}\)	\(8\)
parallelrisch	\(\frac {\sin \left (x \right )}{\ln \left (x \right )}\)	\(8\)
norman	\(\frac {2 \tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \ln \left (x \right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)*ln(x)-sin(x)/x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

sin(x)/ln(x)

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Maxima [A]
time = 0.43, size = 7, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*log(x)-sin(x)/x)/log(x)^2,x, algorithm="maxima")

[Out]

sin(x)/log(x)

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Fricas [A]
time = 0.58, size = 7, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*log(x)-sin(x)/x)/log(x)^2,x, algorithm="fricas")

[Out]

sin(x)/log(x)

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Sympy [A]
time = 0.51, size = 5, normalized size = 0.71 \begin {gather*} \frac {\sin {\left (x \right )}}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*ln(x)-sin(x)/x)/ln(x)**2,x)

[Out]

sin(x)/log(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (7) = 14\).
time = 0.89, size = 20, normalized size = 2.86 \begin {gather*} \frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{\log \left (x\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*log(x)-sin(x)/x)/log(x)^2,x, algorithm="giac")

[Out]

2*tan(1/2*x)/(log(x)*tan(1/2*x)^2 + log(x))

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Mupad [B]
time = 0.28, size = 7, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)*log(x) - sin(x)/x)/log(x)^2,x)

[Out]

sin(x)/log(x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 2.71 \begin {gather*} -\frac {\cos \left (x \right )}{\ln \left (x \right )}+\frac {\sin \left (x \right )}{x \ln \left (x \right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((cos(x)*ln(x)-sin(x)/x)/ln(x)^2,x)

[Out]

-cos(x)/ln(x)+sin(x)/x/ln(x)

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