\(\int \cot ^2(x) (a+b \cot ^2(x))^{3/2} \, dx\) [27]

Optimal result

Integrand size = 17, antiderivative size = 127 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)} \]

[Out]

(a-b)^(3/2)*arctan(cot(x)*(a-b)^(1/2)/(a+b*cot(x)^2)^(1/2))-1/8*(3*a^2-12*a*b+8*b^2)*arctanh(cot(x)*b^(1/2)/(a
+b*cot(x)^2)^(1/2))/b^(1/2)-1/8*(5*a-4*b)*cot(x)*(a+b*cot(x)^2)^(1/2)-1/4*b*cot(x)^3*(a+b*cot(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3751, 488, 596, 537, 223, 212, 385, 209} \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{8 \sqrt {b}}+(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)} \]

[In]

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

[Out]

(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*C
ot[x])/Sqrt[a + b*Cot[x]^2]])/(8*Sqrt[b]) - ((5*a - 4*b)*Cot[x]*Sqrt[a + b*Cot[x]^2])/8 - (b*Cot[x]^3*Sqrt[a +
b*Cot[x]^2])/4

Rule 209

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

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

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

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

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^2 \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (a (4 a-3 b)+(5 a-4 b) b x^2\right )}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}+\frac {\text {Subst}\left (\int \frac {a (5 a-4 b) b-b \left (3 a^2-12 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{8 b} \\ & = -\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}+(a-b)^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )+\frac {1}{8} \left (-3 a^2+12 a b-8 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}+(a-b)^2 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+\frac {1}{8} \left (-3 a^2+12 a b-8 b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right ) \\ & = (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.99 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\frac {\sqrt {-a-b+(a-b) \cos (2 x)} \csc (x) \left (8 \sqrt {2} (a-b)^2 \sqrt {-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {a-b} \left (-\sqrt {2} \left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {-b} \sqrt {-a-b+(a-b) \cos (2 x)} \cot (x) \csc (x) \left (5 a-6 b+2 b \csc ^2(x)\right )\right )\right )}{8 \sqrt {2} \sqrt {a-b} \sqrt {-b} \sqrt {-\left ((-a-b+(a-b) \cos (2 x)) \csc ^2(x)\right )}} \]

[In]

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

[Out]

(Sqrt[-a - b + (a - b)*Cos[2*x]]*Csc[x]*(8*Sqrt[2]*(a - b)^2*Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqr
t[-a - b + (a - b)*Cos[2*x]]] + Sqrt[a - b]*(-(Sqrt[2]*(3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[
x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]]*Cot[x]*Csc[x]*(5*a - 6*b + 2*
b*Csc[x]^2))))/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-b]*Sqrt[-((-a - b + (a - b)*Cos[2*x])*Csc[x]^2)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(285\) vs. \(2(105)=210\).

Time = 0.03 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.25

method	result	size
derivativedivides	\(-\frac {\cot \left (x \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{8}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{8 \sqrt {b}}-b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )+\frac {b \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{2}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{2}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\)	\(286\)
default	\(-\frac {\cot \left (x \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{8}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{8 \sqrt {b}}-b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )+\frac {b \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{2}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{2}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\)	\(286\)

[In]

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

[Out]

-1/4*cot(x)*(a+b*cot(x)^2)^(3/2)-3/8*a*cot(x)*(a+b*cot(x)^2)^(1/2)-3/8*a^2/b^(1/2)*ln(b^(1/2)*cot(x)+(a+b*cot(
x)^2)^(1/2))-b^(3/2)*ln(b^(1/2)*cot(x)+(a+b*cot(x)^2)^(1/2))+1/2*b*cot(x)*(a+b*cot(x)^2)^(1/2)+3/2*b^(1/2)*a*l
n(b^(1/2)*cot(x)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^
2)^(1/2)*cot(x))-2*a/b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))
+a^2*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (105) = 210\).

Time = 0.33 (sec) , antiderivative size = 1134, normalized size of antiderivative = 8.93 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/16*(8*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*co
s(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*co
s(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) -
a - 2*b)/(cos(2*x) - 1))*sin(2*x) + 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a
- b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*((3*a^2 - 12*a*b + 8*b^2 - (3*a^2 -
12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b
*cos(2*x) + b))*sin(2*x) - 4*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a +
b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b - 6*b^2)
*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/
16*(16*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos
(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)
*cos(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x
) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt((
(a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*(8*(a*b - b^2 - (a*b - b^2)*cos(2
*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x
) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*s
qrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b
- 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*
x))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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