3.8 \(\int \frac{\csc ^3(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=49 \[ \frac{a}{8 (a \cos (x)+a)^2}-\frac{1}{8 (a-a \cos (x))}+\frac{1}{4 (a \cos (x)+a)}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a} \]

[Out]

(-3*ArcTanh[Cos[x]])/(8*a) - 1/(8*(a - a*Cos[x])) + a/(8*(a + a*Cos[x])^2) + 1/(4*(a + a*Cos[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0735655, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2667, 44, 206} \[ \frac{a}{8 (a \cos (x)+a)^2}-\frac{1}{8 (a-a \cos (x))}+\frac{1}{4 (a \cos (x)+a)}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + a*Cos[x]),x]

[Out]

(-3*ArcTanh[Cos[x]])/(8*a) - 1/(8*(a - a*Cos[x])) + a/(8*(a + a*Cos[x])^2) + 1/(4*(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 ^3(x)}{a+a \cos (x)} \, dx &=-\left (a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^3} \, dx,x,a \cos (x)\right )\right )\\ &=-\left (a^3 \operatorname{Subst}\left (\int \left (\frac{1}{8 a^3 (a-x)^2}+\frac{1}{4 a^2 (a+x)^3}+\frac{1}{4 a^3 (a+x)^2}+\frac{3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \cos (x)\right )\right )\\ &=-\frac{1}{8 (a-a \cos (x))}+\frac{a}{8 (a+a \cos (x))^2}+\frac{1}{4 (a+a \cos (x))}-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{1}{8 (a-a \cos (x))}+\frac{a}{8 (a+a \cos (x))^2}+\frac{1}{4 (a+a \cos (x))}\\ \end{align*}

Mathematica [A]  time = 0.101573, size = 60, normalized size = 1.22 \[ \frac{-2 \cot ^2\left (\frac{x}{2}\right )+\sec ^2\left (\frac{x}{2}\right )-12 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )+4}{16 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + a*Cos[x]),x]

[Out]

(4 - 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(16*a*(1 + Cos[x]))

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 55, normalized size = 1.1 \begin{align*}{\frac{1}{8\,a \left ( -1+\cos \left ( x \right ) \right ) }}+{\frac{3\,\ln \left ( -1+\cos \left ( x \right ) \right ) }{16\,a}}+{\frac{1}{8\,a \left ( \cos \left ( x \right ) +1 \right ) ^{2}}}+{\frac{1}{4\,a \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{3\,\ln \left ( \cos \left ( x \right ) +1 \right ) }{16\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3/(a+a*cos(x)),x)

[Out]

1/8/a/(-1+cos(x))+3/16/a*ln(-1+cos(x))+1/8/a/(cos(x)+1)^2+1/4/a/(cos(x)+1)-3/16*ln(cos(x)+1)/a

________________________________________________________________________________________

Maxima [A]  time = 1.18647, size = 78, normalized size = 1.59 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) - 2}{8 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} - \frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="maxima")

[Out]

1/8*(3*cos(x)^2 + 3*cos(x) - 2)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a) - 3/16*log(cos(x) + 1)/a + 3/16*log(c
os(x) - 1)/a

________________________________________________________________________________________

Fricas [A]  time = 1.62889, size = 267, normalized size = 5.45 \begin{align*} \frac{6 \, \cos \left (x\right )^{2} - 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 6 \, \cos \left (x\right ) - 4}{16 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 - cos(x) - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + cos(x)^2 - cos(x
) - 1)*log(-1/2*cos(x) + 1/2) + 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{3}{\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)**3/(a+a*cos(x)),x)

[Out]

Integral(csc(x)**3/(cos(x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.12665, size = 70, normalized size = 1.43 \begin{align*} -\frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) - 2}{8 \, a{\left (\cos \left (x\right ) + 1\right )}^{2}{\left (\cos \left (x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="giac")

[Out]

-3/16*log(cos(x) + 1)/a + 3/16*log(-cos(x) + 1)/a + 1/8*(3*cos(x)^2 + 3*cos(x) - 2)/(a*(cos(x) + 1)^2*(cos(x)
- 1))