Integrand size = 15, antiderivative size = 69 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-(a-b) \sqrt {a+b \cot ^2(x)}-\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2} \] 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)
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {1}{3} \sqrt {a+b \cot ^2(x)} \left (4 a-3 b+b \cot ^2(x)\right ) \] Input:
Integrate[Cot[x]*(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*Cot[ x]^2]*(4*a - 3*b + b*Cot[x]^2))/3
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 25, 4153, 25, 353, 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.
\(\displaystyle \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right ) \left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right ) \left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int -\frac {\cot (x) \left (a+b \cot ^2(x)\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot (x) \left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 353 |
\(\displaystyle -\frac {1}{2} \int \frac {\left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot ^2(x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-(a-b) \int \frac {\sqrt {b \cot ^2(x)+a}}{\cot ^2(x)+1}d\cot ^2(x)-\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-(a-b) \left ((a-b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)+2 \sqrt {a+b \cot ^2(x)}\right )-\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-(a-b) \left (\frac {2 (a-b) \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}+2 \sqrt {a+b \cot ^2(x)}\right )-\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-(a-b) \left (2 \sqrt {a+b \cot ^2(x)}-2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )\right )-\frac {2}{3} \left (a+b \cot ^2(x)\right )^{3/2}\right )\) |
Input:
Int[Cot[x]*(a + b*Cot[x]^2)^(3/2),x]
Output:
((-2*(a + b*Cot[x]^2)^(3/2))/3 - (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_)^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_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
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. \(135\) vs. \(2(57)=114\).
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.97
method | result | size |
derivativedivides | \(-\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+b \sqrt {a +b \cot \left (x \right )^{2}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(136\) |
default | \(-\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+b \sqrt {a +b \cot \left (x \right )^{2}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(136\) |
Input:
int(cot(x)*(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-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. 141 vs. \(2 (57) = 114\).
Time = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.78 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left [-\frac {3 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) + 8 \, {\left (2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \, {\left (\cos \left (2 \, x\right ) - 1\right )}}, \frac {3 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) - 4 \, {\left (2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (\cos \left (2 \, x\right ) - 1\right )}}\right ] \] Input:
integrate(cot(x)*(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
[-1/12*(3*((a - b)*cos(2*x) - a + b)*sqrt(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)) + 8*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1), 1/6*(3*((a - b)*cos(2*x) - a + b) *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)) - 4*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1)]
\[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\int \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \cot {\left (x \right )}\, dx \] Input:
integrate(cot(x)*(a+b*cot(x)**2)**(3/2),x)
Output:
Integral((a + b*cot(x)**2)**(3/2)*cot(x), x)
Exception generated. \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(x)*(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. 211 vs. \(2 (57) = 114\).
Time = 0.49 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.06 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{6} \, {\left (3 \, {\left (a - b\right )}^{\frac {3}{2}} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) - \frac {8 \, {\left (3 \, {\left (a b - b^{2}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{4} \sqrt {a - b} - 3 \, {\left (a b^{2} - b^{3}\right )} {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt {a - b} + 2 \, {\left (a b^{3} - b^{4}\right )} \sqrt {a - b}\right )}}{{\left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{3}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate(cot(x)*(a+b*cot(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/6*(3*(a - b)^(3/2)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x) ^2 + b))^2) - 8*(3*(a*b - b^2)*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*s in(x)^2 + b))^4*sqrt(a - b) - 3*(a*b^2 - b^3)*(sqrt(a - b)*sin(x) - sqrt(a *sin(x)^2 - b*sin(x)^2 + b))^2*sqrt(a - b) + 2*(a*b^3 - b^4)*sqrt(a - b))/ ((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - b)^3)*sgn(si n(x))
Time = 11.98 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\mathrm {atanh}\left (\frac {{\left (a-b\right )}^{3/2}\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}{3}-\left (a-b\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \] Input:
int(cot(x)*(a + b*cot(x)^2)^(3/2),x)
Output:
atanh(((a - b)^(3/2)*(a + b*cot(x)^2)^(1/2))/(a^2 - 2*a*b + b^2))*(a - b)^ (3/2) - (a + b*cot(x)^2)^(3/2)/3 - (a - b)*(a + b*cot(x)^2)^(1/2)
\[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\frac {-\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{2} b^{2}-3 \sqrt {\cot \left (x \right )^{2} b +a}\, a^{2}+2 \sqrt {\cot \left (x \right )^{2} b +a}\, a b -3 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) a^{2} b +6 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) a \,b^{2}-3 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) b^{3}}{3 b} \] Input:
int(cot(x)*(a+b*cot(x)^2)^(3/2),x)
Output:
( - sqrt(cot(x)**2*b + a)*cot(x)**2*b**2 - 3*sqrt(cot(x)**2*b + a)*a**2 + 2*sqrt(cot(x)**2*b + a)*a*b - 3*int((sqrt(cot(x)**2*b + a)*cot(x)**3)/(cot (x)**2*b + a),x)*a**2*b + 6*int((sqrt(cot(x)**2*b + a)*cot(x)**3)/(cot(x)* *2*b + a),x)*a*b**2 - 3*int((sqrt(cot(x)**2*b + a)*cot(x)**3)/(cot(x)**2*b + a),x)*b**3)/(3*b)