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

Optimal. Leaf size=11 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

arctanh(sin(d*x+c))/d

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Rubi [A]  time = 0.03, antiderivative size = 11, 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 = {4338, 206} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/d

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 4338

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, 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, Cot] || EqQ[F, cot])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/d

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fricas [B]  time = 0.45, size = 28, normalized size = 2.55 \[ \frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.99, size = 28, normalized size = 2.55 \[ \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(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 = 0.12, size = 12, normalized size = 1.09 \[ \frac {\arctanh \left (\sin \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(sin(d*x+c))/d

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maxima [B]  time = 0.33, size = 26, normalized size = 2.36 \[ \frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(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 = 0.64, size = 15, normalized size = 1.36 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atanh(tan(c/2 + (d*x)/2)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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