### 3.81 $$\int \frac{1}{1-\sin (x)} \, dx$$

Optimal. Leaf size=11 $\frac{\cos (x)}{1-\sin (x)}$

[Out]

Cos[x]/(1 - Sin[x])

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Rubi [A]  time = 0.0078634, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {2648} $\frac{\cos (x)}{1-\sin (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 - Sin[x])^(-1),x]

[Out]

Cos[x]/(1 - Sin[x])

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{1-\sin (x)} \, dx &=\frac{\cos (x)}{1-\sin (x)}\\ \end{align*}

Mathematica [B]  time = 0.0118904, size = 25, normalized size = 2.27 $\frac{2 \sin \left (\frac{x}{2}\right )}{\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 - Sin[x])^(-1),x]

[Out]

(2*Sin[x/2])/(Cos[x/2] - Sin[x/2])

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Maple [A]  time = 0., size = 11, normalized size = 1. \begin{align*} -2\, \left ( -1+\tan \left ( x/2 \right ) \right ) ^{-1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sin(x)),x)

[Out]

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

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Maxima [A]  time = 0.932386, size = 20, normalized size = 1.82 \begin{align*} -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sin(x)/(cos(x) + 1) - 1)

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Fricas [A]  time = 1.94193, size = 61, normalized size = 5.55 \begin{align*} \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(x) + sin(x) + 1)/(cos(x) - sin(x) + 1)

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Sympy [A]  time = 0.380706, size = 8, normalized size = 0.73 \begin{align*} - \frac{2}{\tan{\left (\frac{x}{2} \right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sin(x)),x)

[Out]

-2/(tan(x/2) - 1)

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Giac [A]  time = 1.05628, size = 14, normalized size = 1.27 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(1/2*x) - 1)