### 3.301 $$\int \csc (x) \log (\tan (x)) \sec (x) \, dx$$

Optimal. Leaf size=9 $\frac{1}{2} \log ^2(\tan (x))$

[Out]

Log[Tan[x]]^2/2

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Rubi [A]  time = 0.0232566, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {2620, 29, 6686} $\frac{1}{2} \log ^2(\tan (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \csc (x) \log (\tan (x)) \sec (x) \, dx &=\frac{1}{2} \log ^2(\tan (x))\\ \end{align*}

Mathematica [A]  time = 0.0052715, size = 9, normalized size = 1. $\frac{1}{2} \log ^2(\tan (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

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Maple [A]  time = 0.02, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tan \left ( x \right ) \right ) \right ) ^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(tan(x))/cos(x)/sin(x),x)

[Out]

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

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Maxima [A]  time = 0.932059, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\tan \left (x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(tan(x))/cos(x)/sin(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.03166, size = 35, normalized size = 3.89 \begin{align*} \frac{1}{2} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(tan(x))/cos(x)/sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\tan{\left (x \right )} \right )}}{\sin{\left (x \right )} \cos{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(tan(x))/cos(x)/sin(x),x)

[Out]

Integral(log(tan(x))/(sin(x)*cos(x)), x)

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Giac [A]  time = 1.05607, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\tan \left (x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(tan(x))/cos(x)/sin(x),x, algorithm="giac")

[Out]

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