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

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

[Out]

arctan(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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4338, 203} \[ \frac {\tan ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sin[c + d*x]]/d

Rule 203

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

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Mathematica [A]  time = 0.04, size = 11, normalized size = 1.00 \[ \frac {\tan ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sin[c + d*x]]/d

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fricas [A]  time = 0.49, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

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]

arctan(sin(d*x + c))/d

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giac [A]  time = 0.25, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

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]

arctan(sin(d*x + c))/d

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maple [A]  time = 0.07, size = 13, normalized size = 1.18 \[ -\frac {\arctan \left (\csc \left (d x +c \right )\right )}{d} \]

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]

-1/d*arctan(csc(d*x+c))

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maxima [A]  time = 0.64, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]

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]

arctan(sin(d*x + c))/d

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mupad [B]  time = 0.67, size = 45, normalized size = 4.09 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )-\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{d} \]

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]

(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

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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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