3.193 \(\int \frac {\sec (x)}{\sec (x)+\tan (x)} \, dx\)

Optimal. Leaf size=10 \[ -\frac {\cos (x)}{\sin (x)+1} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3165, 2648} \[ -\frac {\cos (x)}{\sin (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]/(Sec[x] + Tan[x]),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]

Rule 3165

Int[sec[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)])^(m_)
, x_Symbol] :> Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0]
&& IntegerQ[n]

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 23, normalized size = 2.30 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]/(Sec[x] + Tan[x]),x]

[Out]

(2*Sin[x/2])/(Cos[x/2] + Sin[x/2])

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fricas [A]  time = 1.95, size = 18, normalized size = 1.80 \[ -\frac {\cos \relax (x) - \sin \relax (x) + 1}{\cos \relax (x) + \sin \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 10, normalized size = 1.00 \[ -\frac {2}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(tan(1/2*x) + 1)

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maple [A]  time = 0.07, size = 11, normalized size = 1.10 \[ -\frac {2}{\tan \left (\frac {x}{2}\right )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)/(sec(x)+tan(x)),x)

[Out]

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

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maxima [A]  time = 0.43, size = 15, normalized size = 1.50 \[ -\frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sin(x)/(cos(x) + 1) + 1)

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mupad [B]  time = 0.55, size = 10, normalized size = 1.00 \[ -\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)*(tan(x) + 1/cos(x))),x)

[Out]

-2/(tan(x/2) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\relax (x )}}{\tan {\relax (x )} + \sec {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(sec(x)+tan(x)),x)

[Out]

Integral(sec(x)/(tan(x) + sec(x)), x)

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