3.1.7 \(\int \frac {\csc ^2(x)}{a+a \cos (x)} \, dx\) [7]

Optimal. Leaf size=24 \[ -\frac {2 \cot (x)}{3 a}+\frac {\csc (x)}{3 (a+a \cos (x))} \]

[Out]

-2/3*cot(x)/a+1/3*csc(x)/(a+a*cos(x))

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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2751, 3852, 8} \begin {gather*} \frac {\csc (x)}{3 (a \cos (x)+a)}-\frac {2 \cot (x)}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a + a*Cos[x]),x]

[Out]

(-2*Cot[x])/(3*a) + Csc[x]/(3*(a + a*Cos[x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 30, normalized size = 1.25 \begin {gather*} -\frac {(2 \cos (x)+\cos (2 x)) \csc \left (\frac {x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )}{12 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(a + a*Cos[x]),x]

[Out]

-1/12*((2*Cos[x] + Cos[2*x])*Csc[x/2]*Sec[x/2]^3)/a

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Maple [A]
time = 0.07, size = 29, normalized size = 1.21

method	result	size
default	\(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+2 \tan \left (\frac {x}{2}\right )-\frac {1}{\tan \left (\frac {x}{2}\right )}}{4 a}\)	\(29\)
risch	\(-\frac {4 i \left (1+2 \,{\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{i x}+1\right )^{3} a \left ({\mathrm e}^{i x}-1\right )}\)	\(34\)
norman	\(\frac {-\frac {1}{4 a}+\frac {\tan ^{2}\left (\frac {x}{2}\right )}{2 a}+\frac {\tan ^{4}\left (\frac {x}{2}\right )}{12 a}}{\tan \left (\frac {x}{2}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/(a+a*cos(x)),x,method=_RETURNVERBOSE)

[Out]

1/4/a*(1/3*tan(1/2*x)^3+2*tan(1/2*x)-1/tan(1/2*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
time = 0.26, size = 41, normalized size = 1.71 \begin {gather*} \frac {\frac {6 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{12 \, a} - \frac {\cos \left (x\right ) + 1}{4 \, a \sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*sin(x)/(cos(x) + 1) + sin(x)^3/(cos(x) + 1)^3)/a - 1/4*(cos(x) + 1)/(a*sin(x))

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Fricas [A]
time = 0.35, size = 26, normalized size = 1.08 \begin {gather*} -\frac {2 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1}{3 \, {\left (a \cos \left (x\right ) + a\right )} \sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\csc ^{2}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**2/(cos(x) + 1), x)/a

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Giac [A]
time = 0.41, size = 37, normalized size = 1.54 \begin {gather*} \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{12 \, a^{3}} - \frac {1}{4 \, a \tan \left (\frac {1}{2} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a^2*tan(1/2*x)^3 + 6*a^2*tan(1/2*x))/a^3 - 1/4/(a*tan(1/2*x))

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Mupad [B]
time = 0.32, size = 35, normalized size = 1.46 \begin {gather*} \frac {-8\,{\cos \left (\frac {x}{2}\right )}^4+4\,{\cos \left (\frac {x}{2}\right )}^2+1}{12\,a\,{\cos \left (\frac {x}{2}\right )}^3\,\sin \left (\frac {x}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2*(a + a*cos(x))),x)

[Out]

(4*cos(x/2)^2 - 8*cos(x/2)^4 + 1)/(12*a*cos(x/2)^3*sin(x/2))

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