3.9 \(\int \frac{\csc ^4(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=37 \[ -\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)} \]

[Out]

(-4*Cot[x])/(5*a) - (4*Cot[x]^3)/(15*a) + Csc[x]^3/(5*(a + a*Cos[x]))

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Rubi [A]  time = 0.0484188, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2672, 3767} \[ -\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4/(a + a*Cos[x]),x]

[Out]

(-4*Cot[x])/(5*a) - (4*Cot[x]^3)/(15*a) + Csc[x]^3/(5*(a + a*Cos[x]))

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[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 3767

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 ^4(x)}{a+a \cos (x)} \, dx &=\frac{\csc ^3(x)}{5 (a+a \cos (x))}+\frac{4 \int \csc ^4(x) \, dx}{5 a}\\ &=\frac{\csc ^3(x)}{5 (a+a \cos (x))}-\frac{4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{5 a}\\ &=-\frac{4 \cot (x)}{5 a}-\frac{4 \cot ^3(x)}{15 a}+\frac{\csc ^3(x)}{5 (a+a \cos (x))}\\ \end{align*}

Mathematica [A]  time = 0.0565487, size = 38, normalized size = 1.03 \[ \frac{(-6 \cos (x)-2 \cos (2 x)+2 \cos (3 x)+\cos (4 x)) \csc ^3(x)}{15 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4/(a + a*Cos[x]),x]

[Out]

((-6*Cos[x] - 2*Cos[2*x] + 2*Cos[3*x] + Cos[4*x])*Csc[x]^3)/(15*a*(1 + Cos[x]))

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Maple [A]  time = 0.051, size = 45, normalized size = 1.2 \begin{align*}{\frac{1}{16\,a} \left ({\frac{1}{5} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{4}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+6\,\tan \left ( x/2 \right ) -4\, \left ( \tan \left ( x/2 \right ) \right ) ^{-1}-{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^4/(a+a*cos(x)),x)

[Out]

1/16/a*(1/5*tan(1/2*x)^5+4/3*tan(1/2*x)^3+6*tan(1/2*x)-4/tan(1/2*x)-1/3/tan(1/2*x)^3)

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Maxima [B]  time = 1.07647, size = 95, normalized size = 2.57 \begin{align*} \frac{\frac{90 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{20 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{240 \, a} - \frac{{\left (\frac{12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{3}}{48 \, a \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*cos(x)),x, algorithm="maxima")

[Out]

1/240*(90*sin(x)/(cos(x) + 1) + 20*sin(x)^3/(cos(x) + 1)^3 + 3*sin(x)^5/(cos(x) + 1)^5)/a - 1/48*(12*sin(x)^2/
(cos(x) + 1)^2 + 1)*(cos(x) + 1)^3/(a*sin(x)^3)

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Fricas [A]  time = 1.54182, size = 153, normalized size = 4.14 \begin{align*} -\frac{8 \, \cos \left (x\right )^{4} + 8 \, \cos \left (x\right )^{3} - 12 \, \cos \left (x\right )^{2} - 12 \, \cos \left (x\right ) + 3}{15 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*cos(x)),x, algorithm="fricas")

[Out]

-1/15*(8*cos(x)^4 + 8*cos(x)^3 - 12*cos(x)^2 - 12*cos(x) + 3)/((a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)*sin(x)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{4}{\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)**4/(a+a*cos(x)),x)

[Out]

Integral(csc(x)**4/(cos(x) + 1), x)/a

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Giac [A]  time = 1.14274, size = 80, normalized size = 2.16 \begin{align*} -\frac{12 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}{48 \, a \tan \left (\frac{1}{2} \, x\right )^{3}} + \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{5} + 20 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{3} + 90 \, a^{4} \tan \left (\frac{1}{2} \, x\right )}{240 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^4/(a+a*cos(x)),x, algorithm="giac")

[Out]

-1/48*(12*tan(1/2*x)^2 + 1)/(a*tan(1/2*x)^3) + 1/240*(3*a^4*tan(1/2*x)^5 + 20*a^4*tan(1/2*x)^3 + 90*a^4*tan(1/
2*x))/a^5