3.1.81 \(\int \frac {1}{1-\sin (x)} \, dx\) [81]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {2727} \begin {gather*} \frac {\cos (x)}{1-\sin (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Sin[x])^(-1),x]

[Out]

Cos[x]/(1 - Sin[x])

Rule 2727

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).
time = 0.01, size = 25, normalized size = 2.27 \begin {gather*} \frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \end {gather*}

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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Mathics [A]
time = 1.81, size = 10, normalized size = 0.91 \begin {gather*} \frac {-2}{-1+\text {Tan}\left [\frac {x}{2}\right ]} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(1 - Sin[x]),x]')

[Out]

-2 / (-1 + Tan[x / 2])

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Maple [A]
time = 0.02, size = 11, normalized size = 1.00

method	result	size
default	\(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\)	\(11\)
norman	\(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\)	\(11\)
risch	\(\frac {2}{{\mathrm e}^{i x}-i}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 15, normalized size = 1.36 \begin {gather*} -\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \end {gather*}

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 = 0.34, size = 17, normalized size = 1.55 \begin {gather*} \frac {\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end {gather*}

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 = 0.18, size = 8, normalized size = 0.73 \begin {gather*} - \frac {2}{\tan {\left (\frac {x}{2} \right )} - 1} \end {gather*}

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 = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\frac {2}{\tan \left (\frac {x}{2}\right )-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(1/2*x) - 1)

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Mupad [B]
time = 0.03, size = 10, normalized size = 0.91 \begin {gather*} -\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sin(x) - 1),x)

[Out]

-2/(tan(x/2) - 1)

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