\(\int \cot (x) (a+b \cot ^4(x))^{3/2} \, dx\) [61]

Optimal result

Integrand size = 15, antiderivative size = 126 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3751, 1262, 749, 829, 858, 223, 212, 739} \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)} \]

[In]

Int[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]

[Out]

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 749

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 2*p + 1))), x] + Dist[2*(p/(e*(m + 2*p + 1))), Int[(d + e*x)^m*Simp[a*e - c*d*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 829

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

Rule 858

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

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

Mathematica [A] (verified)

Time = 4.81 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.33 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{12} \left (6 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+6 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)} \left (8 a+6 b-3 b \cot ^2(x)+2 b \cot ^4(x)\right )+\frac {3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \cot ^4(x)}}{\sqrt {1+\frac {b \cot ^4(x)}{a}}}\right ) \]

[In]

Integrate[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]

[Out]

(6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + 6*(a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)
/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Cot[x]^4]*(8*a + 6*b - 3*b*Cot[x]^2 + 2*b*Cot[x]^4) + (3*Sqr
t[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Cot[x]^2)/Sqrt[a]]*Sqrt[a + b*Cot[x]^4])/Sqrt[1 + (b*Cot[x]^4)/a])/12

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(103)=206\).

Time = 0.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.94

method	result	size
derivativedivides	\(\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2}-\frac {b^{2} \left (\frac {\cot \left (x \right )^{4} \sqrt {a +b \cot \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \cot \left (x \right )^{4}}}{3 b^{2}}\right )}{2}-\frac {b \sqrt {a +b \cot \left (x \right )^{4}}}{2}+\sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )+\frac {b^{2} \left (\frac {\cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}-a \sqrt {a +b \cot \left (x \right )^{4}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}\)	\(245\)
default	\(\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2}-\frac {b^{2} \left (\frac {\cot \left (x \right )^{4} \sqrt {a +b \cot \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \cot \left (x \right )^{4}}}{3 b^{2}}\right )}{2}-\frac {b \sqrt {a +b \cot \left (x \right )^{4}}}{2}+\sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )+\frac {b^{2} \left (\frac {\cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}-a \sqrt {a +b \cot \left (x \right )^{4}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}\)	\(245\)

[In]

int(cot(x)*(a+b*cot(x)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*b^(3/2)*ln(b^(1/2)*cot(x)^2+(a+b*cot(x)^4)^(1/2))-1/2*b^2*(1/3*cot(x)^4/b*(a+b*cot(x)^4)^(1/2)-2/3*a/b^2*(
a+b*cot(x)^4)^(1/2))-1/2*b*(a+b*cot(x)^4)^(1/2)+b^(1/2)*a*ln(b^(1/2)*cot(x)^2+(a+b*cot(x)^4)^(1/2))+1/2*b^2*(1
/2*cot(x)^2/b*(a+b*cot(x)^4)^(1/2)-1/2*a/b^(3/2)*ln(b^(1/2)*cot(x)^2+(a+b*cot(x)^4)^(1/2)))-a*(a+b*cot(x)^4)^(
1/2)+1/2*(a^2+2*a*b+b^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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (104) = 208\).

Time = 0.48 (sec) , antiderivative size = 1486, normalized size of antiderivative = 11.79 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="fricas")

[Out]

[1/24*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(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)) + 3*((3*a + 2*b)*cos(2*x)^2
- 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(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)) - 2*(a - 2*b)*cos(2*x) + a + 2*b
)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*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)))/(cos(2*x)^2 - 2*cos(2*x) + 1), 1/12
*(3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-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))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + 3*((a +
b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(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)) - ((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*co
s(2*x) + 8*a + 5*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)))/(co
s(2*x)^2 - 2*cos(2*x) + 1), -1/24*(12*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-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)))
- 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(
2*x)^2 - 1)*sqrt(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)) - 2*
(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) + 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*
x) + 8*a + 5*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)))/(cos(2*
x)^2 - 2*cos(2*x) + 1), -1/12*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-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))) - 3*(
(3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-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))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + ((8*a + 11*b)
*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*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)))/(cos(2*x)^2 - 2*cos(2*x) + 1)]

Sympy [F]

\[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\int \left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \cot {\left (x \right )}\, dx \]

[In]

integrate(cot(x)*(a+b*cot(x)**4)**(3/2),x)

[Out]

Integral((a + b*cot(x)**4)**(3/2)*cot(x), x)

Maxima [F]

\[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\int { {\left (b \cot \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \cot \left (x\right ) \,d x } \]

[In]

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cot(x)^4 + a)^(3/2)*cot(x), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (104) = 208\).

Time = 0.54 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.53 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=-\frac {{\left (3 \, a b + 2 \, b^{2}\right )} \arctan \left (-\frac {\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right )}{2 \, \sqrt {a + b}} - \frac {3 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{5} {\left (5 \, a b + 6 \, b^{2}\right )} + 8 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{3} b^{3} - 12 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{4} {\left (a b + 3 \, b^{2}\right )} \sqrt {a + b} + 12 \, {\left (a b^{2} + b^{3}\right )} {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt {a + b} + 3 \, {\left (3 \, a b^{3} + 2 \, b^{4}\right )} {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} - 8 \, {\left (a b^{3} + b^{4}\right )} \sqrt {a + b}}{6 \, {\left ({\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{3}} \]

[In]

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="giac")

[Out]

-1/2*(3*a*b + 2*b^2)*arctan(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))/sqrt(-b
))/sqrt(-b) - 1/2*(a^2 + 2*a*b + b^2)*log(abs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(
x)^2 + b))*(a + b) + sqrt(a + b)*b))/sqrt(a + b) - 1/6*(3*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4
- 2*b*sin(x)^2 + b))^5*(5*a*b + 6*b^2) + 8*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^
2 + b))^3*b^3 - 12*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^4*(a*b + 3*b^2)*s
qrt(a + b) + 12*(a*b^2 + b^3)*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^2*sqrt
(a + b) + 3*(3*a*b^3 + 2*b^4)*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b)) - 8*(a
*b^3 + b^4)*sqrt(a + b))/((sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^2 - b)^3

Mupad [F(-1)]

Timed out. \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\int \mathrm {cot}\left (x\right )\,{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{3/2} \,d x \]

[In]

int(cot(x)*(a + b*cot(x)^4)^(3/2),x)

[Out]

int(cot(x)*(a + b*cot(x)^4)^(3/2), x)