3.210 \(\int \frac {\cos (x)}{-\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]

[Out]

cos(x)+ln(1-cos(x))

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Rubi [A]  time = 0.07, antiderivative size = 10, 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 = {4392, 2833, 43} \[ \cos (x)+\log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(-Cot[x] + Csc[x]),x]

[Out]

Cos[x] + Log[1 - Cos[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 4392

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 2.00 \[ 2 \log \left (\sin \left (\frac {x}{2}\right )\right )-2 \sin ^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(-Cot[x] + Csc[x]),x]

[Out]

2*Log[Sin[x/2]] - 2*Sin[x/2]^2

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fricas [A]  time = 0.80, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(-cot(x)+csc(x)),x, algorithm="fricas")

[Out]

cos(x) + log(-1/2*cos(x) + 1/2)

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giac [A]  time = 0.20, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(-cot(x)+csc(x)),x, algorithm="giac")

[Out]

cos(x) + log(-cos(x) + 1)

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maple [A]  time = 0.10, size = 9, normalized size = 0.90 \[ \cos \relax (x )+\ln \left (\cos \relax (x )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(-cot(x)+csc(x)),x)

[Out]

cos(x)+ln(cos(x)-1)

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maxima [B]  time = 0.65, size = 46, normalized size = 4.60 \[ \frac {2}{\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1} + 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) - \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(-cot(x)+csc(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.62, size = 31, normalized size = 3.10 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(x)/(cot(x) - 1/sin(x)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\cos {\relax (x )}}{\cot {\relax (x )} - \csc {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(-cot(x)+csc(x)),x)

[Out]

-Integral(cos(x)/(cot(x) - csc(x)), x)

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