3.6 \(\int \frac{\csc (x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{2 (a \cos (x)+a)}-\frac{\tanh ^{-1}(\cos (x))}{2 a} \]

[Out]

-ArcTanh[Cos[x]]/(2*a) + 1/(2*(a + a*Cos[x]))

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Rubi [A]  time = 0.0495032, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2667, 44, 206} \[ \frac{1}{2 (a \cos (x)+a)}-\frac{\tanh ^{-1}(\cos (x))}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a + a*Cos[x]),x]

[Out]

-ArcTanh[Cos[x]]/(2*a) + 1/(2*(a + a*Cos[x]))

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 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+a \cos (x)} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^2} \, dx,x,a \cos (x)\right )\right )\\ &=-\left (a \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a+x)^2}+\frac{1}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,a \cos (x)\right )\right )\\ &=\frac{1}{2 (a+a \cos (x))}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{1}{2 (a+a \cos (x))}\\ \end{align*}

Mathematica [A]  time = 0.0313792, size = 42, normalized size = 1.83 \[ \frac{1-2 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{2 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a + a*Cos[x]),x]

[Out]

(1 - 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(2*a*(1 + Cos[x]))

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Maple [A]  time = 0.046, size = 33, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{4\,a}}+{\frac{1}{2\,a \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(a+a*cos(x)),x)

[Out]

1/4/a*ln(-1+cos(x))+1/2/a/(cos(x)+1)-1/4*ln(cos(x)+1)/a

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Maxima [A]  time = 1.13465, size = 42, normalized size = 1.83 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{4 \, a} + \frac{1}{2 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(cos(x) + 1)/a + 1/4*log(cos(x) - 1)/a + 1/2/(a*cos(x) + a)

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Fricas [A]  time = 1.66671, size = 135, normalized size = 5.87 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2}{4 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((cos(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + 1)*log(-1/2*cos(x) + 1/2) - 2)/(a*cos(x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+a*cos(x)),x)

[Out]

Integral(csc(x)/(cos(x) + 1), x)/a

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Giac [A]  time = 1.11544, size = 46, normalized size = 2. \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{1}{2 \, a{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(cos(x) + 1)/a + 1/4*log(-cos(x) + 1)/a + 1/2/(a*(cos(x) + 1))