\(\int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx\) [189]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=-\log (1+\sin (x))+\sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4476, 2912, 45} \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=\sin (x)-\log (\sin (x)+1) \]

[In]

Int[Sin[x]/(Sec[x] + Tan[x]),x]

[Out]

-Log[1 + Sin[x]] + Sin[x]

Rule 45

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 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 4476

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.90 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=-2 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\sin (x) \]

[In]

Integrate[Sin[x]/(Sec[x] + Tan[x]),x]

[Out]

-2*Log[Cos[x/2] + Sin[x/2]] + Sin[x]

Maple [A] (verified)

Time = 1.51 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(-\ln \left (1+\sin \left (x \right )\right )+\sin \left (x \right )\)	\(11\)
default	\(-\ln \left (1+\sin \left (x \right )\right )+\sin \left (x \right )\)	\(11\)
risch	\(i x -\frac {i {\mathrm e}^{i x}}{2}+\frac {i {\mathrm e}^{-i x}}{2}-2 \ln \left (i+{\mathrm e}^{i x}\right )\)	\(33\)

[In]

int(sin(x)/(sec(x)+tan(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=-\log \left (\sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \]

[In]

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

[Out]

-log(sin(x) + 1) + sin(x)

Sympy [F]

\[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=\int \frac {\sin {\left (x \right )}}{\tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(sin(x)/(tan(x) + sec(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (10) = 20\).

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 5.40 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=\frac {2 \, \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (x\right ) + 1\right )}} - 2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=-\log \left (\sin \left (x\right ) + 1\right ) + \sin \left (x\right ) \]

[In]

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

[Out]

-log(sin(x) + 1) + sin(x)

Mupad [B] (verification not implemented)

Time = 22.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 2.10 \[ \int \frac {\sin (x)}{\sec (x)+\tan (x)} \, dx=\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )+\sin \left (x\right ) \]

[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)