### 3.102 $$\int \csc (x) \, dx$$

Optimal. Leaf size=5 $-\tanh ^{-1}(\cos (x))$

[Out]

-ArcTanh[Cos[x]]

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Rubi [A]  time = 0.0022161, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3770} $-\tanh ^{-1}(\cos (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[x],x]

[Out]

-ArcTanh[Cos[x]]

Rule 3770

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

Rubi steps

\begin{align*} \int \csc (x) \, dx &=-\tanh ^{-1}(\cos (x))\\ \end{align*}

Mathematica [B]  time = 0.0025744, size = 17, normalized size = 3.4 $\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csc[x],x]

[Out]

-Log[Cos[x/2]] + Log[Sin[x/2]]

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Maple [A]  time = 0.002, size = 9, normalized size = 1.8 \begin{align*} -\ln \left ( \csc \left ( x \right ) +\cot \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x),x)

[Out]

-ln(csc(x)+cot(x))

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Maxima [A]  time = 0.930988, size = 11, normalized size = 2.2 \begin{align*} -\log \left (\cot \left (x\right ) + \csc \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x),x, algorithm="maxima")

[Out]

-log(cot(x) + csc(x))

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Fricas [B]  time = 2.15113, size = 77, normalized size = 15.4 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.093246, size = 15, normalized size = 3. \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x),x)

[Out]

log(cos(x) - 1)/2 - log(cos(x) + 1)/2

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Giac [A]  time = 1.07252, size = 8, normalized size = 1.6 \begin{align*} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x),x, algorithm="giac")

[Out]

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