\(\int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx\) [219]

Optimal result

Integrand size = 22, antiderivative size = 11 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]

[Out]

arctan(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4423, 209} \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]

[In]

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

[Out]

ArcTan[Sin[c + d*x]]/d

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 {\arctan (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]

[In]

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

[Out]

ArcTan[Sin[c + d*x]]/d

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\arctan \left (\sin \left (d x +c \right )\right )}{d}\)	\(12\)
default	\(\frac {\arctan \left (\sin \left (d x +c \right )\right )}{d}\)	\(12\)
risch	\(-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\)	\(60\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

[In]

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

[Out]

arctan(sin(d*x + c))/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 [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

[In]

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

[Out]

arctan(sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

[In]

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

[Out]

arctan(sin(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 23.94 (sec) , antiderivative size = 45, normalized size of antiderivative = 4.09 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )-\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{d} \]

[In]

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

[Out]

(atan((5*tan(c/2 + (d*x)/2))/2 + tan(c/2 + (d*x)/2)^3/2) - atan(tan(c/2 + (d*x)/2)/2))/d