3.196 \(\int \frac {\sin (x)}{\sec (x)-\tan (x)} \, dx\)

Optimal. Leaf size=14 \[ -\sin (x)-\log (1-\sin (x)) \]

[Out]

-ln(1-sin(x))-sin(x)

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Rubi [A]  time = 0.08, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4391, 2833, 43} \[ -\sin (x)-\log (1-\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(Sec[x] - Tan[x]),x]

[Out]

-Log[1 - Sin[x]] - Sin[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 4391

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.64 \[ -\sin (x)-2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(Sec[x] - Tan[x]),x]

[Out]

-2*Log[Cos[x/2] - Sin[x/2]] - Sin[x]

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fricas [A]  time = 1.46, size = 14, normalized size = 1.00 \[ -\log \left (-\sin \relax (x) + 1\right ) - \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-sin(x) + 1) - sin(x)

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giac [A]  time = 2.39, size = 14, normalized size = 1.00 \[ -\log \left (-\sin \relax (x) + 1\right ) - \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-sin(x) + 1) - sin(x)

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maple [A]  time = 0.10, size = 13, normalized size = 0.93 \[ -\sin \relax (x )-\ln \left (\sin \relax (x )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(sec(x)-tan(x)),x)

[Out]

-sin(x)-ln(sin(x)-1)

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maxima [B]  time = 1.03, size = 54, normalized size = 3.86 \[ -\frac {2 \, \sin \relax (x)}{{\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )} {\left (\cos \relax (x) + 1\right )}} - 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) + \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.61, size = 23, normalized size = 1.64 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )-\sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sin(x)/(tan(x) - 1/cos(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(sec(x)-tan(x)),x)

[Out]

Integral(sin(x)/(-tan(x) + sec(x)), x)

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