3.41 \(\int x \csc ^2(x) \, dx\)

Optimal. Leaf size=9 \[ \log (\sin (x))-x \cot (x) \]

[Out]

-x*cot(x)+ln(sin(x))

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4184, 3475} \[ \log (\sin (x))-x \cot (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Csc[x]^2,x]

[Out]

-(x*Cot[x]) + Log[Sin[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 4184

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 9, normalized size = 1.00 \[ \log (\sin (x))-x \cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Csc[x]^2,x]

[Out]

-(x*Cot[x]) + Log[Sin[x]]

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fricas [B]  time = 0.42, size = 20, normalized size = 2.22 \[ -\frac {x \cos \relax (x) - \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x)}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(x)^2,x, algorithm="fricas")

[Out]

-(x*cos(x) - log(1/2*sin(x))*sin(x))/sin(x)

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giac [B]  time = 0.96, size = 52, normalized size = 5.78 \[ \frac {x \tan \left (\frac {1}{2} \, x\right )^{2} + \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right ) - x}{2 \, \tan \left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(x)^2,x, algorithm="giac")

[Out]

1/2*(x*tan(1/2*x)^2 + log(16*tan(1/2*x)^2/(tan(1/2*x)^4 + 2*tan(1/2*x)^2 + 1))*tan(1/2*x) - x)/tan(1/2*x)

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maple [A]  time = 0.01, size = 10, normalized size = 1.11 \[ -x \cot \relax (x )+\ln \left (\sin \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*csc(x)^2,x)

[Out]

-x*cot(x)+ln(sin(x))

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maxima [B]  time = 0.45, size = 104, normalized size = 11.56 \[ \frac {{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 4 \, x \sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(x)^2,x, algorithm="maxima")

[Out]

1/2*((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*cos(x) + 1) - 4*x*sin(2*x))/(cos(2*x)^2 + sin(2*x)^2 - 2*co
s(2*x) + 1)

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mupad [B]  time = 0.15, size = 9, normalized size = 1.00 \[ \ln \left (\sin \relax (x)\right )-x\,\mathrm {cot}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sin(x)^2,x)

[Out]

log(sin(x)) - x*cot(x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \csc ^{2}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*csc(x)**2,x)

[Out]

Integral(x*csc(x)**2, x)

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