3.1.7 \(\int \cot ^2(x) (x-\tan (x)) \, dx\) [7]

Optimal. Leaf size=13 \[ -\frac {x^2}{2}-x \cot (x) \]

[Out]

-1/2*x^2-x*cot(x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6874, 3556, 3801, 30} \begin {gather*} -\frac {x^2}{2}-x \cot (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*(x - Tan[x]),x]

[Out]

-1/2*x^2 - x*Cot[x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3556

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

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \cot ^2(x) (x-\tan (x)) \, dx &=\int \left (-\cot (x)+x \cot ^2(x)\right ) \, dx\\ &=-\int \cot (x) \, dx+\int x \cot ^2(x) \, dx\\ &=-x \cot (x)-\log (\sin (x))-\int x \, dx+\int \cot (x) \, dx\\ &=-\frac {x^2}{2}-x \cot (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} -\frac {x^2}{2}-x \cot (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2*(x - Tan[x]),x]

[Out]

-1/2*x^2 - x*Cot[x]

________________________________________________________________________________________

Mathics [A]
time = 1.68, size = 11, normalized size = 0.85 \begin {gather*} -\frac {x \left (x+\frac {2}{\text {Tan}\left [x\right ]}\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(x - Tan[x])/Tan[x]^2,x]')

[Out]

-x (x + 2 / Tan[x]) / 2

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 17, normalized size = 1.31

method	result	size
norman	\(\frac {-x -\frac {x^{2} \tan \left (x \right )}{2}}{\tan \left (x \right )}\)	\(17\)
risch	\(-\frac {x^{2}}{2}-i x -\frac {2 i x}{{\mathrm e}^{2 i x}-1}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-tan(x))/tan(x)^2,x,method=_RETURNVERBOSE)

[Out]

(-x-1/2*x^2*tan(x))/tan(x)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (11) = 22\).
time = 0.26, size = 144, normalized size = 11.08 \begin {gather*} -\frac {x^{2} \cos \left (2 \, x\right )^{2} + x^{2} \sin \left (2 \, x\right )^{2} - 2 \, x^{2} \cos \left (2 \, x\right ) + x^{2} - {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, x \sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} - \log \left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-tan(x))/tan(x)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 16, normalized size = 1.23 \begin {gather*} -\frac {x^{2} \tan \left (x\right ) + 2 \, x}{2 \, \tan \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-tan(x))/tan(x)^2,x, algorithm="fricas")

[Out]

-1/2*(x^2*tan(x) + 2*x)/tan(x)

________________________________________________________________________________________

Sympy [A]
time = 0.08, size = 10, normalized size = 0.77 \begin {gather*} - \frac {x^{2}}{2} - \frac {x}{\tan {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-tan(x))/tan(x)**2,x)

[Out]

-x**2/2 - x/tan(x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 18, normalized size = 1.38 \begin {gather*} \frac {-x^{2} \tan x-2 x}{2 \tan x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-tan(x))/tan(x)^2,x)

[Out]

-1/2*(x^2*tan(x) + 2*x)/tan(x)

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 11, normalized size = 0.85 \begin {gather*} -x\,\mathrm {cot}\left (x\right )-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - tan(x))/tan(x)^2,x)

[Out]

- x*cot(x) - x^2/2

________________________________________________________________________________________