Integrand size = 17, antiderivative size = 66 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \cot ^2(x)}-\frac {\left (a+b \cot ^2(x)\right )^{3/2}}{3 b} \] Output:
-(a-b)^(1/2)*arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))+(a+b*cot(x)^2)^(1/2 )-1/3*(a+b*cot(x)^2)^(3/2)/b
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\sqrt {a+b \cot ^2(x)} \left (a-3 b+b \cot ^2(x)\right )}{3 b} \] Input:
Integrate[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]
Output:
-(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 = 0.31 (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 |
| \(\displaystyle \frac {1}{2} \left (-2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {2 \left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+2 \sqrt {a+b \cot ^2(x)}\right )\) |
Input:
Int[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]
Output:
(-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
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 = 0.33 (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 | \(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}+\sqrt {a +b \cot \left (x \right )^{2}}-\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\) | \(84\) |
Input:
int(cot(x)^3*(a+b*cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-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 = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 5.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\left [\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - 4 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}, -\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}\right ] \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm="fricas")
Output:
[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)]
\[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\int \sqrt {a + b \cot ^{2}{\left (x \right )}} \cot ^{3}{\left (x \right )}\, dx \] Input:
integrate(cot(x)**3*(a+b*cot(x)**2)**(1/2),x)
Output:
Integral(sqrt(a + b*cot(x)**2)*cot(x)**3, x)
Exception generated. \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),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. 198 vs. \(2 (54) = 108\).
Time = 0.50 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\frac {1}{6} \, {\left (3 \, \sqrt {a - b} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{4} \sqrt {a - b} {\left (a - 2 \, b\right )} + 6 \, {\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt {a - b} b^{2} + {\left (a b^{2} - 4 \, b^{3}\right )} \sqrt {a - b}\right )}}{{\left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{3}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm="giac")
Output:
1/6*(3*sqrt(a - b)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2) + 4*(3*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^4 *sqrt(a - b)*(a - 2*b) + 6*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x )^2 + b))^2*sqrt(a - b)*b^2 + (a*b^2 - 4*b^3)*sqrt(a - b))/((sqrt(a - b)*s in(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - b)^3)*sgn(sin(x))
Time = 12.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}{3\,b}+2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,\sqrt {\frac {b}{4}-\frac {a}{4}}}{a-b}\right )\,\sqrt {\frac {b}{4}-\frac {a}{4}} \] Input:
int(cot(x)^3*(a + b*cot(x)^2)^(1/2),x)
Output:
(a + b*cot(x)^2)^(1/2) - (a + b*cot(x)^2)^(3/2)/(3*b) + 2*atan((2*(a + b*c ot(x)^2)^(1/2)*(b/4 - a/4)^(1/2))/(a - b))*(b/4 - a/4)^(1/2)
\[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\frac {-\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{2} b +2 \sqrt {\cot \left (x \right )^{2} b +a}\, a +3 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) a b -3 \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )^{3}}{\cot \left (x \right )^{2} b +a}d x \right ) b^{2}}{3 b} \] Input:
int(cot(x)^3*(a+b*cot(x)^2)^(1/2),x)
Output:
( - sqrt(cot(x)**2*b + a)*cot(x)**2*b + 2*sqrt(cot(x)**2*b + a)*a + 3*int( (sqrt(cot(x)**2*b + a)*cot(x)**3)/(cot(x)**2*b + a),x)*a*b - 3*int((sqrt(c ot(x)**2*b + a)*cot(x)**3)/(cot(x)**2*b + a),x)*b**2)/(3*b)