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]

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

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Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {\log (\cos (c+d x))}{d} \]

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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fricas [A]  time = 0.46, size = 14, normalized size = 1.17 \[ -\frac {\log \left (-\cos \left (d x + c\right )\right )}{d} \]

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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giac [B]  time = 0.20, size = 26, normalized size = 2.17 \[ -\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]

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

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maple [A]  time = 0.06, size = 13, normalized size = 1.08 \[ -\frac {\ln \left (\cos \left (d x +c \right )\right )}{d} \]

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 = 0.34, size = 24, normalized size = 2.00 \[ -\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]

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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mupad [B]  time = 0.06, size = 14, normalized size = 1.17 \[ -\frac {\ln \left ({\cos \left (c+d\,x\right )}^2\right )}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(c + d*x)/(sin(c + d*x) - 1/sin(c + d*x)),x)

[Out]

-log(cos(c + d*x)^2)/(2*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

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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