Integrand size = 17, antiderivative size = 59 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \] Output:
-arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)+a/(a-b)/b/(a+b*cot( x)^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \] Input:
Integrate[Cot[x]^3/(a + b*Cot[x]^2)^(3/2),x]
Output:
-(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2)) + a/((a - b)*b* Sqrt[a + b*Cot[x]^2])
Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 25, 4153, 25, 354, 87, 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.
\(\displaystyle \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )^3}{\left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )^3}{\left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int -\frac {\cot ^3(x)}{\left (\cot ^2(x)+1\right ) \left (a+b \cot ^2(x)\right )^{3/2}}d\cot (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot ^3(x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{3/2}}d\cot (x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{2} \int \frac {\cot ^2(x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{3/2}}d\cot ^2(x)\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)}{a-b}+\frac {2 a}{b (a-b) \sqrt {a+b \cot ^2(x)}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b (a-b)}+\frac {2 a}{b (a-b) \sqrt {a+b \cot ^2(x)}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {2 a}{b (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}\right )\) |
Input:
Int[Cot[x]^3/(a + b*Cot[x]^2)^(3/2),x]
Output:
((-2*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]])/(a - b)^(3/2) + (2*a)/((a - b)*b*Sqrt[a + b*Cot[x]^2]))/2
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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {1}{b \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}\) | \(68\) |
default | \(\frac {1}{b \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}\) | \(68\) |
Input:
int(cot(x)^3/(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/b/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(-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. 204 vs. \(2 (51) = 102\).
Time = 0.10 (sec) , antiderivative size = 403, normalized size of antiderivative = 6.83 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\left [-\frac {{\left (a b + b^{2} - {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) - 2 \, {\left (a^{2} - a b - {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left (a b + b^{2} - {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}\right ) - {\left (a^{2} - a b - {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )}\right ] \] Input:
integrate(cot(x)^3/(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
[-1/2*((a*b + b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*log(-sqrt(a - b)*sqr t(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos( 2*x) + a) - 2*(a^2 - a*b - (a^2 - a*b)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^ 2 + 3*a*b^3 - b^4)*cos(2*x)), -((a*b + b^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 - b)) - (a^2 - a*b - (a^2 - a*b)*cos (2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(2*x))]
\[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(x)**3/(a+b*cot(x)**2)**(3/2),x)
Output:
Integral(cot(x)**3/(a + b*cot(x)**2)**(3/2), x)
Exception generated. \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] 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. 109 vs. \(2 (51) = 102\).
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {a - b} a - \sqrt {a - b} b\right )}} + \frac {\frac {a \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a b - b^{2}\right )}} + \frac {\log \left ({\left | -\sqrt {a - b} \sin \left (x\right ) + \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} \right |}\right )}{{\left (a - b\right )}^{\frac {3}{2}}}}{\mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:
integrate(cot(x)^3/(a+b*cot(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/2*log(abs(b))*sgn(sin(x))/(sqrt(a - b)*a - sqrt(a - b)*b) + (a*sin(x)/( sqrt(a*sin(x)^2 - b*sin(x)^2 + b)*(a*b - b^2)) + log(abs(-sqrt(a - b)*sin( x) + sqrt(a*sin(x)^2 - b*sin(x)^2 + b)))/(a - b)^(3/2))/sgn(sin(x))
Time = 11.56 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {a}{\left (a\,b-b^2\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{\sqrt {a-b}}\right )}{{\left (a-b\right )}^{3/2}} \] Input:
int(cot(x)^3/(a + b*cot(x)^2)^(3/2),x)
Output:
a/((a*b - b^2)*(a + b*cot(x)^2)^(1/2)) - atanh((a + b*cot(x)^2)^(1/2)/(a - b)^(1/2))/(a - b)^(3/2)
\[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {-\cot \left (x \right )^{2} \left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )}{\cot \left (x \right )^{4} b^{2}+2 \cot \left (x \right )^{2} a b +a^{2}}d x \right ) b^{2}+\sqrt {\cot \left (x \right )^{2} b +a}-\left (\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \cot \left (x \right )}{\cot \left (x \right )^{4} b^{2}+2 \cot \left (x \right )^{2} a b +a^{2}}d x \right ) a b}{b \left (\cot \left (x \right )^{2} b +a \right )} \] Input:
int(cot(x)^3/(a+b*cot(x)^2)^(3/2),x)
Output:
( - cot(x)**2*int((sqrt(cot(x)**2*b + a)*cot(x))/(cot(x)**4*b**2 + 2*cot(x )**2*a*b + a**2),x)*b**2 + sqrt(cot(x)**2*b + a) - int((sqrt(cot(x)**2*b + a)*cot(x))/(cot(x)**4*b**2 + 2*cot(x)**2*a*b + a**2),x)*a*b)/(b*(cot(x)** 2*b + a))