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

Optimal. Leaf size=22 \[ -\frac {\tanh ^{-1}(\cos (x))}{2 a}-\frac {\cot (x) \csc (x)}{2 a} \]

[Out]

-1/2*arctanh(cos(x))/a-1/2*cot(x)*csc(x)/a

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3175, 3768, 3770} \[ -\frac {\tanh ^{-1}(\cos (x))}{2 a}-\frac {\cot (x) \csc (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a - a*Cos[x]^2),x]

[Out]

-ArcTanh[Cos[x]]/(2*a) - (Cot[x]*Csc[x])/(2*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 (x)}{a-a \cos ^2(x)} \, dx &=\frac {\int \csc ^3(x) \, dx}{a}\\ &=-\frac {\cot (x) \csc (x)}{2 a}+\frac {\int \csc (x) \, dx}{2 a}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{2 a}-\frac {\cot (x) \csc (x)}{2 a}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 51, normalized size = 2.32 \[ \frac {-\frac {1}{8} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{8} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{2} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{2} \log \left (\cos \left (\frac {x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a - a*Cos[x]^2),x]

[Out]

(-1/8*Csc[x/2]^2 - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2 + Sec[x/2]^2/8)/a

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fricas [B]  time = 1.45, size = 48, normalized size = 2.18 \[ -\frac {{\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2 \, \cos \relax (x)}{4 \, {\left (a \cos \relax (x)^{2} - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 38, normalized size = 1.73 \[ -\frac {\log \left (\cos \relax (x) + 1\right )}{4 \, a} + \frac {\log \left (-\cos \relax (x) + 1\right )}{4 \, a} + \frac {\cos \relax (x)}{2 \, {\left (\cos \relax (x)^{2} - 1\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.10, size = 44, normalized size = 2.00 \[ \frac {1}{4 a \left (-1+\cos \relax (x )\right )}+\frac {\ln \left (-1+\cos \relax (x )\right )}{4 a}+\frac {1}{4 a \left (\cos \relax (x )+1\right )}-\frac {\ln \left (\cos \relax (x )+1\right )}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(a-a*cos(x)^2),x)

[Out]

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

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maxima [B]  time = 0.48, size = 37, normalized size = 1.68 \[ \frac {\cos \relax (x)}{2 \, {\left (a \cos \relax (x)^{2} - a\right )}} - \frac {\log \left (\cos \relax (x) + 1\right )}{4 \, a} + \frac {\log \left (\cos \relax (x) - 1\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 26, normalized size = 1.18 \[ -\frac {\cos \relax (x)}{2\,\left (a-a\,{\cos \relax (x)}^2\right )}-\frac {\mathrm {atanh}\left (\cos \relax (x)\right )}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)*(a - a*cos(x)^2)),x)

[Out]

- cos(x)/(2*(a - a*cos(x)^2)) - atanh(cos(x))/(2*a)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\csc {\relax (x )}}{\cos ^{2}{\relax (x )} - 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a-a*cos(x)**2),x)

[Out]

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

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