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[c + d*x]]/d

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Rubi [A]  time = 0.0264199, antiderivative size = 11, 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 = {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 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])

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])

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*}

Mathematica [A]  time = 0.0016969, size = 11, normalized size = 1. \[ \frac{\tanh ^{-1}(\sin (c+d x))}{d} \]

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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Maple [A]  time = 0.075, size = 12, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

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 = 1.07318, size = 35, normalized size = 3.18 \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(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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Fricas [B]  time = 0.489127, size = 76, normalized size = 6.91 \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(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\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(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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Giac [B]  time = 1.18535, size = 38, normalized size = 3.45 \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(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