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

Optimal. Leaf size=44 \[ \frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \csc (x)+a}-\frac{3 \cot (x) \csc (x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0659157, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3818, 3787, 3767, 8, 3768, 3770} \[ \frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \csc (x)+a}-\frac{3 \cot (x) \csc (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4/(a + a*Csc[x]),x]

[Out]

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

Rule 3818

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(d^2*Cot[e
+ f*x]*(d*Csc[e + f*x])^(n - 2))/(f*(a + b*Csc[e + f*x])), x] - Dist[d^2/(a*b), Int[(d*Csc[e + f*x])^(n - 2)*(
b*(n - 2) - a*(n - 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

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]

Rule 8

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

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

Mathematica [A]  time = 0.324768, size = 83, normalized size = 1.89 \[ \frac{-4 \tan \left (\frac{x}{2}\right )+4 \cot \left (\frac{x}{2}\right )-\csc ^2\left (\frac{x}{2}\right )+\sec ^2\left (\frac{x}{2}\right )+12 \log \left (\sin \left (\frac{x}{2}\right )\right )-12 \log \left (\cos \left (\frac{x}{2}\right )\right )-\frac{16 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{8 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4/(a + a*Csc[x]),x]

[Out]

(4*Cot[x/2] - Csc[x/2]^2 - 12*Log[Cos[x/2]] + 12*Log[Sin[x/2]] + Sec[x/2]^2 - (16*Sin[x/2])/(Cos[x/2] + Sin[x/
2]) - 4*Tan[x/2])/(8*a)

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Maple [A]  time = 0.033, size = 67, normalized size = 1.5 \begin{align*}{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{2\,a}\tan \left ({\frac{x}{2}} \right ) }+2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{3}{2\,a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a*tan(1/2*x)^2-1/2/a*tan(1/2*x)+2/a/(tan(1/2*x)+1)-1/8/a/tan(1/2*x)^2+1/2/a/tan(1/2*x)+3/2/a*ln(tan(1/2*x)
)

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Maxima [B]  time = 0.991286, size = 131, normalized size = 2.98 \begin{align*} -\frac{\frac{4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a} + \frac{\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{20 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{8 \,{\left (\frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac{3 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*sin(x)/(cos(x) + 1) - sin(x)^2/(cos(x) + 1)^2)/a + 1/8*(3*sin(x)/(cos(x) + 1) + 20*sin(x)^2/(cos(x) +
1)^2 - 1)/(a*sin(x)^2/(cos(x) + 1)^2 + a*sin(x)^3/(cos(x) + 1)^3) + 3/2*log(sin(x)/(cos(x) + 1))/a

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Fricas [B]  time = 0.488282, size = 433, normalized size = 9.84 \begin{align*} \frac{8 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )^{2} - 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \,{\left (4 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - 6 \, \cos \left (x\right ) - 4}{4 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) +{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right ) - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*cos(x)^3 + 6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(1/2*cos(x) + 1
/2) + 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(-1/2*cos(x) + 1/2) - 2*(4*cos(x)^2 + co
s(x) - 2)*sin(x) - 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) + (a*cos(x)^2 - a)*sin(x) - a)

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

[Out]

Integral(csc(x)**4/(csc(x) + 1), x)/a

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Giac [A]  time = 1.39513, size = 99, normalized size = 2.25 \begin{align*} \frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{2 \, a} + \frac{a \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, a \tan \left (\frac{1}{2} \, x\right )}{8 \, a^{2}} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} - \frac{18 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac{1}{2} \, x\right ) + 1}{8 \, a \tan \left (\frac{1}{2} \, x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*log(abs(tan(1/2*x)))/a + 1/8*(a*tan(1/2*x)^2 - 4*a*tan(1/2*x))/a^2 + 2/(a*(tan(1/2*x) + 1)) - 1/8*(18*tan(
1/2*x)^2 - 4*tan(1/2*x) + 1)/(a*tan(1/2*x)^2)