3.211 \(\int \frac{\tan (x)}{-\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=5 \[ x+\tanh ^{-1}(\sin (x)) \]

[Out]

x + ArcTanh[Sin[x]]

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Rubi [A]  time = 0.0944992, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4392, 2839, 3770, 8} \[ x+\tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/(-Cot[x] + Csc[x]),x]

[Out]

x + ArcTanh[Sin[x]]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3770

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\tan (x)}{-\cot (x)+\csc (x)} \, dx &=\int \frac{\sin (x) \tan (x)}{1-\cos (x)} \, dx\\ &=\int 1 \, dx+\int \sec (x) \, dx\\ &=x+\tanh ^{-1}(\sin (x))\\ \end{align*}

Mathematica [B]  time = 0.0163073, size = 46, normalized size = 9.2 \[ 2 \left (\frac{x}{2}-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/(-Cot[x] + Csc[x]),x]

[Out]

2*(x/2 - Log[Cos[x/2] - Sin[x/2]]/2 + Log[Cos[x/2] + Sin[x/2]]/2)

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Maple [B]  time = 0.095, size = 19, normalized size = 3.8 \begin{align*} \ln \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) +x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(-cot(x)+csc(x)),x)

[Out]

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

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Maxima [B]  time = 1.67212, size = 53, normalized size = 10.6 \begin{align*} 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(-cot(x)+csc(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.487646, size = 65, normalized size = 13. \begin{align*} x + \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(-cot(x)+csc(x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(-cot(x)+csc(x)),x)

[Out]

-Integral(tan(x)/(cot(x) - csc(x)), x)

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Giac [B]  time = 1.13415, size = 27, normalized size = 5.4 \begin{align*} x + \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(-cot(x)+csc(x)),x, algorithm="giac")

[Out]

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