3.9 \(\int \frac{\csc ^3(x)}{a-a \cos ^2(x)} \, dx\)

Optimal. Leaf size=35 \[ -\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{\cot (x) \csc ^3(x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a} \]

[Out]

(-3*ArcTanh[Cos[x]])/(8*a) - (3*Cot[x]*Csc[x])/(8*a) - (Cot[x]*Csc[x]^3)/(4*a)

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Rubi [A]  time = 0.0551938, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 3768, 3770} \[ -\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{\cot (x) \csc ^3(x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a - a*Cos[x]^2),x]

[Out]

(-3*ArcTanh[Cos[x]])/(8*a) - (3*Cot[x]*Csc[x])/(8*a) - (Cot[x]*Csc[x]^3)/(4*a)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\csc ^3(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \csc ^5(x) \, dx}{a}\\ &=-\frac{\cot (x) \csc ^3(x)}{4 a}+\frac{3 \int \csc ^3(x) \, dx}{4 a}\\ &=-\frac{3 \cot (x) \csc (x)}{8 a}-\frac{\cot (x) \csc ^3(x)}{4 a}+\frac{3 \int \csc (x) \, dx}{8 a}\\ &=-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{3 \cot (x) \csc (x)}{8 a}-\frac{\cot (x) \csc ^3(x)}{4 a}\\ \end{align*}

Mathematica [B]  time = 0.0062478, size = 75, normalized size = 2.14 \[ \frac{-\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{3}{32} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{3}{32} \sec ^2\left (\frac{x}{2}\right )+\frac{3}{8} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{3}{8} \log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a - a*Cos[x]^2),x]

[Out]

((-3*Csc[x/2]^2)/32 - Csc[x/2]^4/64 - (3*Log[Cos[x/2]])/8 + (3*Log[Sin[x/2]])/8 + (3*Sec[x/2]^2)/32 + Sec[x/2]
^4/64)/a

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Maple [B]  time = 0.037, size = 66, normalized size = 1.9 \begin{align*}{\frac{1}{16\,a \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}+{\frac{3}{16\,a \left ( 1+\cos \left ( x \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\cos \left ( x \right ) \right ) }{16\,a}}-{\frac{1}{16\,a \left ( \cos \left ( x \right ) -1 \right ) ^{2}}}+{\frac{3}{16\,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)^2),x)

[Out]

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

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Maxima [A]  time = 0.95155, size = 69, normalized size = 1.97 \begin{align*} \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + 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)^2),x, algorithm="maxima")

[Out]

1/8*(3*cos(x)^3 - 5*cos(x))/(a*cos(x)^4 - 2*a*cos(x)^2 + a) - 3/16*log(cos(x) + 1)/a + 3/16*log(cos(x) - 1)/a

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Fricas [B]  time = 1.97039, size = 232, normalized size = 6.63 \begin{align*} \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a-a*cos(x)^2),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^3 - 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/
2*cos(x) + 1/2) - 10*cos(x))/(a*cos(x)^4 - 2*a*cos(x)^2 + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\csc ^{3}{\left (x \right )}}{\cos ^{2}{\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)**2),x)

[Out]

-Integral(csc(x)**3/(cos(x)**2 - 1), x)/a

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Giac [A]  time = 1.12851, size = 63, normalized size = 1.8 \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 )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a-a*cos(x)^2),x, algorithm="giac")

[Out]

-3/16*log(cos(x) + 1)/a + 3/16*log(-cos(x) + 1)/a + 1/8*(3*cos(x)^3 - 5*cos(x))/((cos(x)^2 - 1)^2*a)