3.288 \(\int \frac{1}{a \cot (x)+b \csc (x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\log (a \cos (x)+b)}{a} \]

[Out]

-(Log[b + a*Cos[x]]/a)

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Rubi [A]  time = 0.0350371, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3160, 2668, 31} \[ -\frac{\log (a \cos (x)+b)}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x] + b*Csc[x])^(-1),x]

[Out]

-(Log[b + a*Cos[x]]/a)

Rule 3160

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a \cot (x)+b \csc (x)} \, dx &=\int \frac{\sin (x)}{b+a \cos (x)} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{b+x} \, dx,x,a \cos (x)\right )}{a}\\ &=-\frac{\log (b+a \cos (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0161616, size = 12, normalized size = 1. \[ -\frac{\log (a \cos (x)+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x] + b*Csc[x])^(-1),x]

[Out]

-(Log[b + a*Cos[x]]/a)

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Maple [A]  time = 0.038, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( b+a\cos \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cot(x)+b*csc(x)),x)

[Out]

-ln(b+a*cos(x))/a

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Maxima [B]  time = 1.4615, size = 61, normalized size = 5.08 \begin{align*} -\frac{\log \left (a + b - \frac{{\left (a - b\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a} + \frac{\log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.06224, size = 30, normalized size = 2.5 \begin{align*} -\frac{\log \left (a \cos \left (x\right ) + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x)),x, algorithm="fricas")

[Out]

-log(a*cos(x) + b)/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a \cot{\left (x \right )} + b \csc{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x)),x)

[Out]

Integral(1/(a*cot(x) + b*csc(x)), x)

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Giac [A]  time = 1.13109, size = 18, normalized size = 1.5 \begin{align*} -\frac{\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cot(x)+b*csc(x)),x, algorithm="giac")

[Out]

-log(abs(a*cos(x) + b))/a