### 3.247 $$\int \frac{1}{\sin (x)+\tan (x)} \, dx$$

Optimal. Leaf size=24 $-\frac{1}{2} \csc ^2(x)-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)$

[Out]

-ArcTanh[Cos[x]]/2 + (Cot[x]*Csc[x])/2 - Csc[x]^2/2

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Rubi [A]  time = 0.0523432, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {4397, 2706, 2606, 30, 2611, 3770} $-\frac{1}{2} \csc ^2(x)-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(Sin[x] + Tan[x])^(-1),x]

[Out]

-ArcTanh[Cos[x]]/2 + (Cot[x]*Csc[x])/2 - Csc[x]^2/2

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

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

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

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 \frac{1}{\sin (x)+\tan (x)} \, dx &=\int \frac{\cot (x)}{1+\cos (x)} \, dx\\ &=-\int \cot ^2(x) \csc (x) \, dx+\int \cot (x) \csc ^2(x) \, dx\\ &=\frac{1}{2} \cot (x) \csc (x)+\frac{1}{2} \int \csc (x) \, dx-\operatorname{Subst}(\int x \, dx,x,\csc (x))\\ &=-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)-\frac{\csc ^2(x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.020161, size = 35, normalized size = 1.46 $-\frac{1}{4} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sin[x] + Tan[x])^(-1),x]

[Out]

-Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2 - Sec[x/2]^2/4

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Maple [A]  time = 0.036, size = 24, normalized size = 1. \begin{align*} -{\frac{1}{2\,\cos \left ( x \right ) +2}}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{4}}+{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)+tan(x)),x)

[Out]

-1/2/(cos(x)+1)-1/4*ln(cos(x)+1)+1/4*ln(cos(x)-1)

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Maxima [A]  time = 0.95352, size = 34, normalized size = 1.42 \begin{align*} -\frac{\sin \left (x\right )^{2}}{4 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{1}{2} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sin(x)+tan(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.31762, size = 132, normalized size = 5.5 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{4 \,{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sin(x)+tan(x)),x, algorithm="fricas")

[Out]

-1/4*((cos(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + 1)*log(-1/2*cos(x) + 1/2) + 2)/(cos(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin{\left (x \right )} + \tan{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sin(x)+tan(x)),x)

[Out]

Integral(1/(sin(x) + tan(x)), x)

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Giac [A]  time = 1.06748, size = 38, normalized size = 1.58 \begin{align*} \frac{\cos \left (x\right ) - 1}{4 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{1}{4} \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sin(x)+tan(x)),x, algorithm="giac")

[Out]

1/4*(cos(x) - 1)/(cos(x) + 1) + 1/4*log(-(cos(x) - 1)/(cos(x) + 1))