3.82 \(\int x \tan ^2(x) \, dx\)

Optimal. Leaf size=15 \[ -\frac{x^2}{2}+x \tan (x)+\log (\cos (x)) \]

[Out]

-x^2/2 + Log[Cos[x]] + x*Tan[x]

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Rubi [A]  time = 0.0143883, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3720, 3475, 30} \[ -\frac{x^2}{2}+x \tan (x)+\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2/2 + Log[Cos[x]] + x*Tan[x]

Rule 3720

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 3475

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

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.0181929, size = 15, normalized size = 1. \[ -\frac{x^2}{2}+x \tan (x)+\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2/2 + Log[Cos[x]] + x*Tan[x]

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Maple [A]  time = 0.009, size = 20, normalized size = 1.3 \begin{align*} x\tan \left ( x \right ) -{\frac{{x}^{2}}{2}}-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.42822, size = 144, normalized size = 9.6 \begin{align*} -\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 (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 2.02897, size = 66, normalized size = 4.4 \begin{align*} -\frac{1}{2} \, x^{2} + x \tan \left (x\right ) + \frac{1}{2} \, \log \left (\frac{1}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.160963, size = 19, normalized size = 1.27 \begin{align*} - \frac{x^{2}}{2} + x \tan{\left (x \right )} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07227, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{2} \, x^{2} + x \tan \left (x\right ) + \frac{1}{2} \, \log \left (\frac{4}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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