3.20 \(\int \frac{\sin (x)}{a+b \cot (x)} \, dx\)

Optimal. Leaf size=66 \[ -\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^2 \tanh ^{-1}\left (\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]

[Out]

(b^2*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) - (b*Sin[x])
/(a^2 + b^2)

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Rubi [A]  time = 0.0868971, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {3511, 3486, 2638, 3509, 206} \[ -\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{b^2 \tanh ^{-1}\left (\frac{\sin (x) (b-a \cot (x))}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + b*Cot[x]),x]

[Out]

(b^2*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) - (b*Sin[x])
/(a^2 + b^2)

Rule 3511

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

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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{\sin (x)}{a+b \cot (x)} \, dx &=\frac{\int (a-b \cot (x)) \sin (x) \, dx}{a^2+b^2}+\frac{b^2 \int \frac{\csc (x)}{a+b \cot (x)} \, dx}{a^2+b^2}\\ &=-\frac{b \sin (x)}{a^2+b^2}+\frac{a \int \sin (x) \, dx}{a^2+b^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,(-b+a \cot (x)) \sin (x)\right )}{a^2+b^2}\\ &=\frac{b^2 \tanh ^{-1}\left (\frac{(b-a \cot (x)) \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{a \cos (x)}{a^2+b^2}-\frac{b \sin (x)}{a^2+b^2}\\ \end{align*}

Mathematica [A]  time = 0.225355, size = 62, normalized size = 0.94 \[ \frac{2 b^2 \tanh ^{-1}\left (\frac{b \tan \left (\frac{x}{2}\right )-a}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{a \cos (x)+b \sin (x)}{a^2+b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + b*Cot[x]),x]

[Out]

(2*b^2*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x] + b*Sin[x])/(a^2 + b^2)

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Maple [A]  time = 0.075, size = 84, normalized size = 1.3 \begin{align*} 2\,{\frac{-\tan \left ( x/2 \right ) b-a}{ \left ({a}^{2}+{b}^{2} \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}+8\,{\frac{{b}^{2}}{ \left ( 4\,{a}^{2}+4\,{b}^{2} \right ) \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(sin(x)/(a+b*cot(x)),x)

[Out]

2/(a^2+b^2)*(-tan(1/2*x)*b-a)/(tan(1/2*x)^2+1)+8*b^2/(4*a^2+4*b^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(sin(x)/(a+b*cot(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.8022, size = 350, normalized size = 5.3 \begin{align*} \frac{\sqrt{a^{2} + b^{2}} b^{2} \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 \,{\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) - 2 \,{\left (a^{2} b + b^{3}\right )} \sin \left (x\right )}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(a^2 + b^2)*b^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)) - 2*(a^3 + a*b^2)*cos(x) - 2*(a^2*b +
b^3)*sin(x))/(a^4 + 2*a^2*b^2 + b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)/(a + b*cot(x)), x)

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Giac [A]  time = 1.31927, size = 127, normalized size = 1.92 \begin{align*} -\frac{b^{2} \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 )}{{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (b \tan \left (\frac{1}{2} \, x\right ) + a\right )}}{{\left (a^{2} + b^{2}\right )}{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2*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)))/(a^2 + b
^2)^(3/2) - 2*(b*tan(1/2*x) + a)/((a^2 + b^2)*(tan(1/2*x)^2 + 1))