\(\int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\) [226]

Optimal result

Integrand size = 24, antiderivative size = 11 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4423, 212} \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/d

Rule 212

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

Rule 4423

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]

[In]

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

[Out]

ArcTanh[Sin[c + d*x]]/d

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\operatorname {arctanh}\left (\sin \left (d x +c \right )\right )}{d}\)	\(12\)
default	\(\frac {\operatorname {arctanh}\left (\sin \left (d x +c \right )\right )}{d}\)	\(12\)
risch	\(\frac {\ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\)	\(37\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.55 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]

[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

Sympy [F]

\[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\int \frac {\cot {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.36 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.55 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 22.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]

[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