Integrand size = 17, antiderivative size = 66 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \cot ^2(x)}-\frac {\left (a+b \cot ^2(x)\right )^{3/2}}{3 b} \]
-1/3*(a+b*cot(x)^2)^(3/2)/b-arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))*(a-b )^(1/2)+(a+b*cot(x)^2)^(1/2)
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\sqrt {a+b \cot ^2(x)} \left (a-3 b+b \cot ^2(x)\right )}{3 b} \]
-(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]) - (Sqrt[a + b*Cot [x]^2]*(a - 3*b + b*Cot[x]^2))/(3*b)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 25, 4153, 25, 354, 90, 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.
| \(\displaystyle \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right )^3 \sqrt {a+b \tan \left (x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right )^3 \sqrt {b \tan \left (x+\frac {\pi }{2}\right )^2+a}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int -\frac {\cot ^3(x) \sqrt {a+b \cot ^2(x)}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot ^3(x) \sqrt {b \cot ^2(x)+a}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {1}{2} \int \frac {\cot ^2(x) \sqrt {b \cot ^2(x)+a}}{\cot ^2(x)+1}d\cot ^2(x)\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\sqrt {b \cot ^2(x)+a}}{\cot ^2(x)+1}d\cot ^2(x)-\frac {2 \left (a+b \cot ^2(x)\right )^{3/2}}{3 b}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left ((a-b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)-\frac {2 \left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+2 \sqrt {a+b \cot ^2(x)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 (a-b) \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}-\frac {2 \left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+2 \sqrt {a+b \cot ^2(x)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {2 \left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+2 \sqrt {a+b \cot ^2(x)}\right )\) |
(-2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] + 2*Sqrt[a + b*C ot[x]^2] - (2*(a + b*Cot[x]^2)^(3/2))/(3*b))/2
3.1.19.3.1 Defintions of rubi rules used
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]))
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}+\sqrt {a +b \cot \left (x \right )^{2}}-\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(84\) |
| default | \(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}+\sqrt {a +b \cot \left (x \right )^{2}}-\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(84\) |
-1/3*(a+b*cot(x)^2)^(3/2)/b+(a+b*cot(x)^2)^(1/2)-b/(-a+b)^(1/2)*arctan((a+ b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+a/(-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. \(2 (54) = 108\).
Time = 0.34 (sec) , antiderivative size = 330, normalized size of antiderivative = 5.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\left [\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - 4 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}, -\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}\right ] \]
[1/12*(3*(b*cos(2*x) - 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)) - 4*((a - 4*b)*cos(2*x) - a + 2*b)*sqrt(((a - b)*cos(2*x) - a - b)/(co s(2*x) - 1)))/(b*cos(2*x) - b), -1/6*(3*(b*cos(2*x) - b)*sqrt(-a + b)*arct an(-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*((a - 4*b)*cos(2*x) - a + 2*b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x) - b)]
\[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\int \sqrt {a + b \cot ^{2}{\left (x \right )}} \cot ^{3}{\left (x \right )}\, dx \]
Exception generated. \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to convert to real sageVARb Error: Bad Argument ValueUnable to convert to real sageVARb Error: Bad Arg ument Val
Time = 15.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}{3\,b}+2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,\sqrt {\frac {b}{4}-\frac {a}{4}}}{a-b}\right )\,\sqrt {\frac {b}{4}-\frac {a}{4}} \]