3.819 \(\int (2 \cot (2 x)-3 \sin (3 x)) \, dx\)

Optimal. Leaf size=10 \[ \cos (3 x)+\log (\sin (2 x)) \]

[Out]

cos(3*x)+ln(sin(2*x))

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3475, 2638} \[ \cos (3 x)+\log (\sin (2 x)) \]

Antiderivative was successfully verified.

[In]

Int[2*Cot[2*x] - 3*Sin[3*x],x]

[Out]

Cos[3*x] + Log[Sin[2*x]]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int (2 \cot (2 x)-3 \sin (3 x)) \, dx &=2 \int \cot (2 x) \, dx-3 \int \sin (3 x) \, dx\\ &=\cos (3 x)+\log (\sin (2 x))\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \cos (3 x)+\log (\sin (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[2*Cot[2*x] - 3*Sin[3*x],x]

[Out]

Cos[3*x] + Log[Sin[2*x]]

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fricas [A]  time = 0.91, size = 18, normalized size = 1.80 \[ 4 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x) + \log \left (-\frac {1}{2} \, \cos \relax (x) \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*cot(2*x)-3*sin(3*x),x, algorithm="fricas")

[Out]

4*cos(x)^3 - 3*cos(x) + log(-1/2*cos(x)*sin(x))

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giac [A]  time = 0.15, size = 11, normalized size = 1.10 \[ \cos \left (3 \, x\right ) + \log \left ({\left | \sin \left (2 \, x\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*cot(2*x)-3*sin(3*x),x, algorithm="giac")

[Out]

cos(3*x) + log(abs(sin(2*x)))

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maple [A]  time = 0.03, size = 17, normalized size = 1.70 \[ -\frac {\ln \left (\cot ^{2}\left (2 x \right )+1\right )}{2}+\cos \left (3 x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*cot(2*x)-3*sin(3*x),x)

[Out]

-1/2*ln(cot(2*x)^2+1)+cos(3*x)

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maxima [A]  time = 0.32, size = 10, normalized size = 1.00 \[ \cos \left (3 \, x\right ) + \log \left (\sin \left (2 \, x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*cot(2*x)-3*sin(3*x),x, algorithm="maxima")

[Out]

cos(3*x) + log(sin(2*x))

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mupad [B]  time = 3.08, size = 24, normalized size = 2.40 \[ \cos \left (3\,x\right )+\ln \left (\cos \left (\frac {x}{2}\right )\,\left (\sin \left (\frac {x}{2}\right )-2\,{\sin \left (\frac {x}{2}\right )}^3\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*cot(2*x) - 3*sin(3*x),x)

[Out]

cos(3*x) + log(cos(x/2)*(sin(x/2) - 2*sin(x/2)^3))

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sympy [A]  time = 0.06, size = 10, normalized size = 1.00 \[ \log {\left (\sin {\left (2 x \right )} \right )} + \cos {\left (3 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*cot(2*x)-3*sin(3*x),x)

[Out]

log(sin(2*x)) + cos(3*x)

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