Integrand size = 17, antiderivative size = 88 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )+(a-b) \sqrt {a+b \cot ^2(x)}+\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b} \] Output:
-(a-b)^(3/2)*arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))+(a-b)*(a+b*cot(x)^2 )^(1/2)+1/3*(a+b*cot(x)^2)^(3/2)-1/5*(a+b*cot(x)^2)^(5/2)/b
Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\sqrt {a+b \cot ^2(x)} \left (3 a^2-20 a b+15 b^2+(6 a-5 b) b \cot ^2(x)+3 b^2 \cot ^4(x)\right )}{15 b} \] Input:
Integrate[Cot[x]^3*(a + b*Cot[x]^2)^(3/2),x]
Output:
-((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]) - (Sqrt[a + b*C ot[x]^2]*(3*a^2 - 20*a*b + 15*b^2 + (6*a - 5*b)*b*Cot[x]^2 + 3*b^2*Cot[x]^ 4))/(15*b)
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {3042, 25, 4153, 25, 354, 90, 60, 60, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right )^3 \left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right )^3 \left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int -\frac {\cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot ^3(x) \left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {1}{2} \int \frac {\cot ^2(x) \left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot ^2(x)\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot ^2(x)-\frac {2 \left (a+b \cot ^2(x)\right )^{5/2}}{5 b}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left ((a-b) \int \frac {\sqrt {b \cot ^2(x)+a}}{\cot ^2(x)+1}d\cot ^2(x)-\frac {2 \left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left ((a-b) \left ((a-b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)+2 \sqrt {a+b \cot ^2(x)}\right )-\frac {2 \left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left ((a-b) \left (\frac {2 (a-b) \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}+2 \sqrt {a+b \cot ^2(x)}\right )-\frac {2 \left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left ((a-b) \left (2 \sqrt {a+b \cot ^2(x)}-2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )\right )-\frac {2 \left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
Input:
Int[Cot[x]^3*(a + b*Cot[x]^2)^(3/2),x]
Output:
((2*(a + b*Cot[x]^2)^(3/2))/3 - (2*(a + b*Cot[x]^2)^(5/2))/(5*b) + (a - b) *(-2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] + 2*Sqrt[a + b* Cot[x]^2]))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs. \(2(72)=144\).
Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.70
| method | result | size |
| derivativedivides | \(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {5}{2}}}{5 b}+\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-b \sqrt {a +b \cot \left (x \right )^{2}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(150\) |
| default | \(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {5}{2}}}{5 b}+\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-b \sqrt {a +b \cot \left (x \right )^{2}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(150\) |
Input:
int(cot(x)^3*(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/5*(a+b*cot(x)^2)^(5/2)/b+1/3*b*cot(x)^2*(a+b*cot(x)^2)^(1/2)+4/3*a*(a+b *cot(x)^2)^(1/2)+b^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2) )-b*(a+b*cot(x)^2)^(1/2)-2*a*b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(- a+b)^(1/2))+a^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (72) = 144\).
Time = 0.13 (sec) , antiderivative size = 486, normalized size of antiderivative = 5.52 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left [-\frac {15 \, {\left ({\left (a b - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a b - b^{2} - 2 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} - 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) + 4 \, {\left ({\left (3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right )} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left (3 \, a^{2} - 20 \, a b + 12 \, b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{60 \, {\left (b \cos \left (2 \, x\right )^{2} - 2 \, b \cos \left (2 \, x\right ) + b\right )}}, -\frac {15 \, {\left ({\left (a b - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a b - b^{2} - 2 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left ({\left (3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right )} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left (3 \, a^{2} - 20 \, a b + 12 \, b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{30 \, {\left (b \cos \left (2 \, x\right )^{2} - 2 \, b \cos \left (2 \, x\right ) + b\right )}}\right ] \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
[-1/60*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 2*(a*b - b^2)*cos(2*x))*s qrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 - 2*((a - b )*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) + 4*((3*a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*b + 13*b^2 - 2*(3*a^2 - 20*a*b + 12*b^2) *cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x)^2 - 2*b*cos(2*x) + b), -1/30*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 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))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) + 2*((3* a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*b + 13*b^2 - 2*(3*a^2 - 2 0*a*b + 12*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))) /(b*cos(2*x)^2 - 2*b*cos(2*x) + b)]
\[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\int \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (x \right )}\, dx \] Input:
integrate(cot(x)**3*(a+b*cot(x)**2)**(3/2),x)
Output:
Integral((a + b*cot(x)**2)**(3/2)*cot(x)**3, x)
Exception generated. \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (72) = 144\).
Time = 1.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.82 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\frac {1}{30} \, {\left (15 \, {\left (a - b\right )}^{\frac {3}{2}} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) + \frac {4 \, {\left (15 \, {\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{8} \sqrt {a - b} + 90 \, {\left (a b^{2} - b^{3}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{6} \sqrt {a - b} + 10 \, {\left (3 \, a^{2} b^{2} - 17 \, a b^{3} + 14 \, b^{4}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{4} \sqrt {a - b} + 70 \, {\left (a b^{4} - b^{5}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt {a - b} + {\left (3 \, a^{2} b^{4} - 26 \, a b^{5} + 23 \, b^{6}\right )} \sqrt {a - b}\right )}}{{\left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{5}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(3/2),x, algorithm="giac")
Output:
1/30*(15*(a - b)^(3/2)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x )^2 + b))^2) + 4*(15*(a^2 - 4*a*b + 3*b^2)*(sqrt(a - b)*sin(x) - sqrt(a*si n(x)^2 - b*sin(x)^2 + b))^8*sqrt(a - b) + 90*(a*b^2 - b^3)*(sqrt(a - b)*si n(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^6*sqrt(a - b) + 10*(3*a^2*b^2 - 17*a*b^3 + 14*b^4)*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b) )^4*sqrt(a - b) + 70*(a*b^4 - b^5)*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2*sqrt(a - b) + (3*a^2*b^4 - 26*a*b^5 + 23*b^6)*sqrt(a - b))/((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - b)^5)*s gn(sin(x))
Time = 19.01 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left (\frac {a}{3\,b}-\frac {a-b}{3\,b}\right )\,{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{5/2}}{5\,b}+\left (a-b\right )\,\left (\frac {a}{b}-\frac {a-b}{b}\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}+\mathrm {atan}\left (\frac {{\left (a-b\right )}^{3/2}\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,1{}\mathrm {i}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i} \] Input:
int(cot(x)^3*(a + b*cot(x)^2)^(3/2),x)
Output:
atan(((a - b)^(3/2)*(a + b*cot(x)^2)^(1/2)*1i)/(a^2 - 2*a*b + b^2))*(a - b )^(3/2)*1i - (a + b*cot(x)^2)^(5/2)/(5*b) + (a/(3*b) - (a - b)/(3*b))*(a + b*cot(x)^2)^(3/2) + (a - b)*(a/b - (a - b)/b)*(a + b*cot(x)^2)^(1/2)
\[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\frac {-3 \sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{4} b^{2}-6 \sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{2} a b +5 \sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{2} b^{2}+12 \sqrt {\cot \left (x \right )^{2} b +a}\, a^{2}-10 \sqrt {\cot \left (x \right )^{2} b +a}\, a b +15 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) a^{2} b -30 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) a \,b^{2}+15 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) b^{3}}{15 b} \] Input:
int(cot(x)^3*(a+b*cot(x)^2)^(3/2),x)
Output:
( - 3*sqrt(cot(x)**2*b + a)*cot(x)**4*b**2 - 6*sqrt(cot(x)**2*b + a)*cot(x )**2*a*b + 5*sqrt(cot(x)**2*b + a)*cot(x)**2*b**2 + 12*sqrt(cot(x)**2*b + a)*a**2 - 10*sqrt(cot(x)**2*b + a)*a*b + 15*int((sqrt(cot(x)**2*b + a)*cot (x)**3)/(cot(x)**2*b + a),x)*a**2*b - 30*int((sqrt(cot(x)**2*b + a)*cot(x) **3)/(cot(x)**2*b + a),x)*a*b**2 + 15*int((sqrt(cot(x)**2*b + a)*cot(x)**3 )/(cot(x)**2*b + a),x)*b**3)/(15*b)