3.19 \(\int \frac{\csc (x)}{a+b \cot (x)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}\left (\frac{\sin (x) (a \cot (x)-b)}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]

[Out]

-(ArcTanh[((-b + a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])

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Rubi [A]  time = 0.0270206, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3509, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sin (x) (a \cot (x)-b)}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a + b*Cot[x]),x]

[Out]

-(ArcTanh[((-b + a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])

Rule 3509

Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Dist[f^(-1), Subst[Int[1/(a^
2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]

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

Mathematica [A]  time = 0.0309696, size = 38, normalized size = 1.06 \[ \frac{2 \tanh ^{-1}\left (\frac{b \tan \left (\frac{x}{2}\right )-a}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a + b*Cot[x]),x]

[Out]

(2*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]

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Maple [A]  time = 0.059, size = 35, normalized size = 1. \begin{align*} 2\,{\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tan \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(a+b*cot(x)),x)

[Out]

2/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tan(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+b*cot(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.72917, size = 242, normalized size = 6.72 \begin{align*} \frac{\log \left (-\frac{2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - 2 \, b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cos \left (x\right ) - b \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}}\right )}{2 \, \sqrt{a^{2} + b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+b*cot(x)),x, algorithm="fricas")

[Out]

1/2*log(-(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 + 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/
(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2))/sqrt(a^2 + b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+b*cot(x)),x)

[Out]

Integral(csc(x)/(a + b*cot(x)), x)

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Giac [A]  time = 1.34669, size = 82, normalized size = 2.28 \begin{align*} -\frac{\log \left (\frac{{\left | 2 \, b \tan \left (\frac{1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac{1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+b*cot(x)),x, algorithm="giac")

[Out]

-log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/sqrt(a^2 + b
^2)