\(\int \frac {\cot (x)}{(a+b \cot ^4(x))^{5/2}} \, dx\) [64]

Optimal result

Integrand size = 15, antiderivative size = 117 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}} \]

[Out]

1/2*arctanh((a-b*cot(x)^2)/(a+b)^(1/2)/(a+b*cot(x)^4)^(1/2))/(a+b)^(5/2)+1/6*(-a-b*cot(x)^2)/a/(a+b)/(a+b*cot(
x)^4)^(3/2)+1/6*(-3*a^2-b*(5*a+2*b)*cot(x)^2)/a^2/(a+b)^2/(a+b*cot(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262, 755, 837, 12, 739, 212} \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}+\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}} \]

[In]

Int[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a + b*Cot[x]^2)/(6*a*(a + b)
*(a + b*Cot[x]^4)^(3/2)) - (3*a^2 + b*(5*a + 2*b)*Cot[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Cot[x]^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot ^2(x)\right )\right ) \\ & = -\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-3 a-2 b-2 b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot ^2(x)\right )}{6 a (a+b)} \\ & = -\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {3 a^2 b}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{6 a^2 b (a+b)^2} \\ & = -\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{2 (a+b)^2} \\ & = -\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^2} \\ & = \frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a^2 (4 a+b)+3 a b (2 a+b) \cot ^2(x)+3 a^2 b \cot ^4(x)+b^2 (5 a+2 b) \cot ^6(x)}{6 a^2 (a+b)^2 \left (a+b \cot ^4(x)\right )^{3/2}} \]

[In]

Integrate[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a^2*(4*a + b) + 3*a*b*(2*a +
b)*Cot[x]^2 + 3*a^2*b*Cot[x]^4 + b^2*(5*a + 2*b)*Cot[x]^6)/(6*a^2*(a + b)^2*(a + b*Cot[x]^4)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(105)=210\).

Time = 0.14 (sec) , antiderivative size = 586, normalized size of antiderivative = 5.01

method	result	size
derivativedivides	\(\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (b +\sqrt {-a b}\right )^{2} \left (-b +\sqrt {-a b}\right )^{2} \sqrt {a +b}}-\frac {\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (-b +\sqrt {-a b}\right ) a}+\frac {-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (b +\sqrt {-a b}\right ) a}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (-b +\sqrt {-a b}\right )^{2} a^{2} \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (b +\sqrt {-a b}\right )^{2} a^{2} \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(586\)
default	\(\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (b +\sqrt {-a b}\right )^{2} \left (-b +\sqrt {-a b}\right )^{2} \sqrt {a +b}}-\frac {\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (-b +\sqrt {-a b}\right ) a}+\frac {-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (b +\sqrt {-a b}\right ) a}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (-b +\sqrt {-a b}\right )^{2} a^{2} \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (b +\sqrt {-a b}\right )^{2} a^{2} \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(586\)

[In]

int(cot(x)/(a+b*cot(x)^4)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (103) = 206\).

Time = 0.52 (sec) , antiderivative size = 1365, normalized size of antiderivative = 11.67 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2 - 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3
*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*c
os(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(
2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 4*((2*a^4 + a^3*
b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 + 5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b -
a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3
- b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 +
5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^
4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^
7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4
*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)), -1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2
- 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqr
t(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)
*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2
- b^2)*cos(2*x))) + 2*((2*a^4 + a^3*b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 +
5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b - a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4
*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3 - b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(
cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b
+ 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^
3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 -
4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x))]

Sympy [F]

\[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(cot(x)/(a+b*cot(x)**4)**(5/2),x)

[Out]

Integral(cot(x)/(a + b*cot(x)**4)**(5/2), x)

Maxima [F]

\[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (b \cot \left (x\right )^{4} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="maxima")

[Out]

integrate(cot(x)/(b*cot(x)^4 + a)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (103) = 206\).

Time = 0.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.36 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=-\frac {{\left (2 \, {\left (\frac {{\left (2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} - b^{4}\right )} \sin \left (x\right )^{2}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}} + \frac {3 \, {\left (3 \, a b^{3} + b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {3 \, {\left (a^{2} b^{2} - 5 \, a b^{3} - 2 \, b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {5 \, a b^{3} + 2 \, b^{4}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}}{6 \, {\left (a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} \sqrt {a + b} + b \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="giac")

[Out]

-1/6*((2*((2*a^3*b - a^2*b^2 - 4*a*b^3 - b^4)*sin(x)^2/(a^4*b + 2*a^3*b^2 + a^2*b^3) + 3*(3*a*b^3 + b^4)/(a^4*
b + 2*a^3*b^2 + a^2*b^3))*sin(x)^2 + 3*(a^2*b^2 - 5*a*b^3 - 2*b^4)/(a^4*b + 2*a^3*b^2 + a^2*b^3))*sin(x)^2 + (
5*a*b^3 + 2*b^4)/(a^4*b + 2*a^3*b^2 + a^2*b^3))/(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b)^(3/2) - 1/2*log(a
bs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))*sqrt(a + b) + b))/((a^2 + 2*a*b
+ b^2)*sqrt(a + b))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{5/2}} \,d x \]

[In]

int(cot(x)/(a + b*cot(x)^4)^(5/2),x)

[Out]

int(cot(x)/(a + b*cot(x)^4)^(5/2), x)