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

Optimal. Leaf size=11 \[ \log (\sin (x))-\log (\sin (x)+1) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4391, 2707, 36, 29, 31} \[ \log (\sin (x))-\log (\sin (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(Sec[x] + Tan[x]),x]

[Out]

Log[Sin[x]] - Log[1 + Sin[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

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

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.82 \[ \log (\sin (x))-2 \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(Sec[x] + Tan[x]),x]

[Out]

-2*Log[Cos[x/2] + Sin[x/2]] + Log[Sin[x]]

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fricas [A]  time = 0.50, size = 13, normalized size = 1.18 \[ \log \left (\frac {1}{2} \, \sin \relax (x)\right ) - \log \left (\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1/2*sin(x)) - log(sin(x) + 1)

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giac [A]  time = 0.16, size = 12, normalized size = 1.09 \[ -\log \left (\sin \relax (x) + 1\right ) + \log \left ({\left | \sin \relax (x) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sin(x) + 1) + log(abs(sin(x)))

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maple [A]  time = 0.12, size = 8, normalized size = 0.73 \[ -\ln \left (1+\csc \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(1+csc(x))

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maxima [B]  time = 0.57, size = 25, normalized size = 2.27 \[ -2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.57, size = 15, normalized size = 1.36 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)*(tan(x) + 1/cos(x))),x)

[Out]

log(tan(x/2)) - 2*log(tan(x/2) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)/(tan(x) + sec(x)), x)

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