Integrand size = 17, antiderivative size = 59 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}-\frac {\cot (x)}{(a-b) \sqrt {a+b \cot ^2(x)}} \] Output:
arctan((a-b)^(1/2)*cot(x)/(a+b*cot(x)^2)^(1/2))/(a-b)^(3/2)-cot(x)/(a-b)/( a+b*cot(x)^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(59)=118\).
Time = 0.48 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.32 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {(-a+b) \cot (x) \sqrt {1+\frac {b \cot ^2(x)}{a}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {-\frac {(a-b) \cot ^2(x)}{a}}}{\sqrt {1+\frac {b \cot ^2(x)}{a}}}\right ) (-a-b+(a-b) \cos (2 x)) \sqrt {-\frac {(a-b) \cot ^2(x)}{a}} \csc (x) \sec (x)}{(a-b)^2 \sqrt {a+b \cot ^2(x)} \sqrt {1+\frac {b \cot ^2(x)}{a}}} \] Input:
Integrate[Cot[x]^2/(a + b*Cot[x]^2)^(3/2),x]
Output:
((-a + b)*Cot[x]*Sqrt[1 + (b*Cot[x]^2)/a] + (ArcTanh[Sqrt[-(((a - b)*Cot[x ]^2)/a)]/Sqrt[1 + (b*Cot[x]^2)/a]]*(-a - b + (a - b)*Cos[2*x])*Sqrt[-(((a - b)*Cot[x]^2)/a)]*Csc[x]*Sec[x])/2)/((a - b)^2*Sqrt[a + b*Cot[x]^2]*Sqrt[ 1 + (b*Cot[x]^2)/a])
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 4153, 373, 291, 216}
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.
\(\displaystyle \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (x+\frac {\pi }{2}\right )^2}{\left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -\int \frac {\cot ^2(x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{3/2}}d\cot (x)\) |
\(\Big \downarrow \) 373 |
\(\displaystyle \frac {\int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)}{a-b}-\frac {\cot (x)}{(a-b) \sqrt {a+b \cot ^2(x)}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {(b-a) \cot ^2(x)}{b \cot ^2(x)+a}}d\frac {\cot (x)}{\sqrt {b \cot ^2(x)+a}}}{a-b}-\frac {\cot (x)}{(a-b) \sqrt {a+b \cot ^2(x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}-\frac {\cot (x)}{(a-b) \sqrt {a+b \cot ^2(x)}}\) |
Input:
Int[Cot[x]^2/(a + b*Cot[x]^2)^(3/2),x]
Output:
ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) - Cot[x]/( (a - b)*Sqrt[a + b*Cot[x]^2])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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 = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.68
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right )}{a \sqrt {a +b \cot \left (x \right )^{2}}}-\frac {b \cot \left (x \right )}{\left (a -b \right ) a \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{\left (a -b \right )^{2} b^{2}}\) | \(99\) |
default | \(-\frac {\cot \left (x \right )}{a \sqrt {a +b \cot \left (x \right )^{2}}}-\frac {b \cot \left (x \right )}{\left (a -b \right ) a \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{\left (a -b \right )^{2} b^{2}}\) | \(99\) |
Input:
int(cot(x)^2/(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-cot(x)/a/(a+b*cot(x)^2)^(1/2)-b/(a-b)*cot(x)/a/(a+b*cot(x)^2)^(1/2)+1/(a- b)^2*(b^4*(a-b))^(1/2)/b^2*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^ 2)^(1/2)*cot(x))
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (51) = 102\).
Time = 0.13 (sec) , antiderivative size = 430, normalized size of antiderivative = 7.29 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a - b\right )} \sqrt {-a + b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - b\right )} \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + a^{2} - 2 \, b^{2} + 4 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right ) + 4 \, {\left (a - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{4 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a - b\right )} \sqrt {a - b} \arctan \left (-\frac {{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - b\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - a^{2} + b^{2} - 2 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )}\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (2 \, x\right )\right )}}\right ] \] Input:
integrate(cot(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
[-1/4*(((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 - b)*cos(2*x) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*x)) + 4*(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x)), -1/2*( ((a - b)*cos(2*x) - a - b)*sqrt(a - b)*arctan(-((a - b)*cos(2*x) - b)*sqrt (a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a^2 - 2 *a*b + b^2)*cos(2*x)^2 - a^2 + b^2 - 2*(a*b - b^2)*cos(2*x))) + 2*(a - b)* sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^3 - a^2*b - a *b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x))]
\[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(x)**2/(a+b*cot(x)**2)**(3/2),x)
Output:
Integral(cot(x)**2/(a + b*cot(x)**2)**(3/2), x)
Exception generated. \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(x)^2/(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(b-a>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (51) = 102\).
Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {{\left (\sqrt {b} \log \left ({\left | -\sqrt {-a + b} + \sqrt {b} \right |}\right ) + \sqrt {-a + b}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{a \sqrt {-a + b} \sqrt {b} - \sqrt {-a + b} b^{\frac {3}{2}}} + \frac {\frac {\sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a} \cos \left (x\right )}{{\left (a \cos \left (x\right )^{2} - b \cos \left (x\right )^{2} - a\right )} {\left (a - b\right )}} - \frac {\log \left ({\left | -\sqrt {-a + b} \cos \left (x\right ) + \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a} \right |}\right )}{{\left (a - b\right )} \sqrt {-a + b}}}{\mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:
integrate(cot(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="giac")
Output:
(sqrt(b)*log(abs(-sqrt(-a + b) + sqrt(b))) + sqrt(-a + b))*sgn(sin(x))/(a* sqrt(-a + b)*sqrt(b) - sqrt(-a + b)*b^(3/2)) + (sqrt(-a*cos(x)^2 + b*cos(x )^2 + a)*cos(x)/((a*cos(x)^2 - b*cos(x)^2 - a)*(a - b)) - log(abs(-sqrt(-a + b)*cos(x) + sqrt(-a*cos(x)^2 + b*cos(x)^2 + a)))/((a - b)*sqrt(-a + b)) )/sgn(sin(x))
Timed out. \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (x\right )}^2}{{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}} \,d x \] Input:
int(cot(x)^2/(a + b*cot(x)^2)^(3/2),x)
Output:
int(cot(x)^2/(a + b*cot(x)^2)^(3/2), x)
\[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {-\cot \left (x \right )^{2} \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}}{\cot \left (x \right )^{4} b^{2}+2 \cot \left (x \right )^{2} a b +a^{2}}d x \right ) a b -\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )-\left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}}{\cot \left (x \right )^{4} b^{2}+2 \cot \left (x \right )^{2} a b +a^{2}}d x \right ) a^{2}}{a \left (\cot \left (x \right )^{2} b +a \right )} \] Input:
int(cot(x)^2/(a+b*cot(x)^2)^(3/2),x)
Output:
( - (cot(x)**2*int(sqrt(cot(x)**2*b + a)/(cot(x)**4*b**2 + 2*cot(x)**2*a*b + a**2),x)*a*b + sqrt(cot(x)**2*b + a)*cot(x) + int(sqrt(cot(x)**2*b + a) /(cot(x)**4*b**2 + 2*cot(x)**2*a*b + a**2),x)*a**2))/(a*(cot(x)**2*b + a))