3.3.1 \(\int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx\) [201]

Optimal. Leaf size=13 \[ -\log (1-\sin (x))+\log (\sin (x)) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4476, 2786, 36, 29, 31} \begin {gather*} \log (\sin (x))-\log (1-\sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(Sec[x] - Tan[x]),x]

[Out]

-Log[1 - Sin[x]] + Log[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 2786

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

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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.69 \begin {gather*} -2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log (\sin (x)) \end {gather*}

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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Maple [A]
time = 0.15, size = 8, normalized size = 0.62

method	result	size
derivativedivides	\(-\ln \left (-1+\csc \left (x \right )\right )\)	\(8\)
default	\(-\ln \left (-1+\csc \left (x \right )\right )\)	\(8\)
risch	\(-2 \ln \left ({\mathrm e}^{i x}-i\right )+\ln \left ({\mathrm e}^{2 i x}-1\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(sec(x)-tan(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(-1+csc(x))

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Maxima [A]
time = 0.28, size = 25, normalized size = 1.92 \begin {gather*} -2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}

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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Fricas [A]
time = 3.09, size = 15, normalized size = 1.15 \begin {gather*} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (x \right )}}{- \tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \end {gather*}

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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Giac [A]
time = 0.40, size = 14, normalized size = 1.08 \begin {gather*} -\log \left (-\sin \left (x\right ) + 1\right ) + \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end {gather*}

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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Mupad [B]
time = 0.56, size = 15, normalized size = 1.15 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right ) \end {gather*}

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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