3.224 \(\int \frac{\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\log (\cos (c+d x))}{d} \]

[Out]

-(Log[Cos[c + d*x]]/d)

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Rubi [A]  time = 0.0306763, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4334, 260} \[ -\frac{\log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/d)

Rule 4334

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 260

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

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{\log (\cos (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.007376, size = 12, normalized size = 1. \[ -\frac{\log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/d)

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Maple [A]  time = 0.041, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

-ln(cos(d*x+c))/d

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Maxima [A]  time = 1.11038, size = 32, normalized size = 2.67 \begin{align*} -\frac{\log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1))/d

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Fricas [A]  time = 0.487465, size = 31, normalized size = 2.58 \begin{align*} -\frac{\log \left (-\cos \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="fricas")

[Out]

-log(-cos(d*x + c))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (c + d x \right )}}{- \sin{\left (c + d x \right )} + \csc{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)/(-sin(c + d*x) + csc(c + d*x)), x)

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Giac [B]  time = 1.15391, size = 35, normalized size = 2.92 \begin{align*} -\frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2*(log(abs(sin(d*x + c) + 1)) + log(abs(sin(d*x + c) - 1)))/d