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] + Log[Sin[2*x]]

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Rubi [A]  time = 0.0076479, antiderivative size = 10, normalized size of antiderivative = 1., 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 3475

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

Rule 2638

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

Mathematica [A]  time = 0.0102941, size = 10, normalized size = 1. \[ \cos (3 x)+\log (\sin (2 x)) \]

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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Maple [A]  time = 0.005, size = 17, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( \left ( \cot \left ( 2\,x \right ) \right ) ^{2}+1 \right ) }{2}}+\cos \left ( 3\,x \right ) \end{align*}

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 = 0.977552, size = 14, normalized size = 1.4 \begin{align*} \cos \left (3 \, x\right ) + \log \left (\sin \left (2 \, x\right )\right ) \end{align*}

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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Fricas [A]  time = 2.46583, size = 66, normalized size = 6.6 \begin{align*} 4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right ) + \log \left (-\frac{1}{2} \, \cos \left (x\right ) \sin \left (x\right )\right ) \end{align*}

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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Sympy [A]  time = 0.065541, size = 10, normalized size = 1. \begin{align*} \log{\left (\sin{\left (2 x \right )} \right )} + \cos{\left (3 x \right )} \end{align*}

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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Giac [A]  time = 1.09397, size = 15, normalized size = 1.5 \begin{align*} \cos \left (3 \, x\right ) + \log \left ({\left | \sin \left (2 \, x\right ) \right |}\right ) \end{align*}

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)))