3.1.9 \(\int \frac {\csc ^4(x)}{a+a \cos (x)} \, dx\) [9]

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

[Out]

-4/5*cot(x)/a-4/15*cot(x)^3/a+1/5*csc(x)^3/(a+a*cos(x))

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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {2751, 3852} \begin {gather*} -\frac {4 \cot ^3(x)}{15 a}-\frac {4 \cot (x)}{5 a}+\frac {\csc ^3(x)}{5 (a \cos (x)+a)} \end {gather*}

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 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 ^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 \text {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*}

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Mathematica [A]
time = 0.07, size = 38, normalized size = 1.03 \begin {gather*} \frac {(-6 \cos (x)-2 \cos (2 x)+2 \cos (3 x)+\cos (4 x)) \csc ^3(x)}{15 a (1+\cos (x))} \end {gather*}

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 = 0.09, size = 45, normalized size = 1.22

method	result	size
default	\(\frac {\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+6 \tan \left (\frac {x}{2}\right )-\frac {4}{\tan \left (\frac {x}{2}\right )}-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}}{16 a}\)	\(45\)
risch	\(\frac {16 i \left (6 \,{\mathrm e}^{3 i x}+2 \,{\mathrm e}^{2 i x}-2 \,{\mathrm e}^{i x}-1\right )}{15 \left ({\mathrm e}^{i x}-1\right )^{3} a \left ({\mathrm e}^{i x}+1\right )^{5}}\)	\(48\)
norman	\(\frac {-\frac {1}{48 a}-\frac {\tan ^{2}\left (\frac {x}{2}\right )}{4 a}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {\tan ^{6}\left (\frac {x}{2}\right )}{12 a}+\frac {\tan ^{8}\left (\frac {x}{2}\right )}{80 a}}{\tan \left (\frac {x}{2}\right )^{3}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (31) = 62\).
time = 0.30, size = 70, normalized size = 1.89 \begin {gather*} \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 {gather*}

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 = 0.35, size = 53, normalized size = 1.43 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\csc ^{4}{\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)**4/(a+a*cos(x)),x)

[Out]

Integral(csc(x)**4/(cos(x) + 1), x)/a

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Giac [A]
time = 0.56, size = 59, normalized size = 1.59 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.37, size = 45, normalized size = 1.22 \begin {gather*} \frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-60\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-5}{240\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^4*(a + a*cos(x))),x)

[Out]

(90*tan(x/2)^4 - 60*tan(x/2)^2 + 20*tan(x/2)^6 + 3*tan(x/2)^8 - 5)/(240*a*tan(x/2)^3)

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