Integrand size = 17, antiderivative size = 60 \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {\csc (x) \sec (x)}{a \sqrt {a \sec ^2(x)}}-\frac {2 \tan (x)}{a \sqrt {a \sec ^2(x)}}+\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \] Output:
-csc(x)*sec(x)/a/(a*sec(x)^2)^(1/2)-2*tan(x)/a/(a*sec(x)^2)^(1/2)+1/3*sin( x)^2*tan(x)/a/(a*sec(x)^2)^(1/2)
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.52 \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\sec ^3(x) \left (-3 \csc (x)-6 \sin (x)+\sin ^3(x)\right )}{3 \left (a \sec ^2(x)\right )^{3/2}} \] Input:
Integrate[Cot[x]^2/(a + a*Tan[x]^2)^(3/2),x]
Output:
(Sec[x]^3*(-3*Csc[x] - 6*Sin[x] + Sin[x]^3))/(3*(a*Sec[x]^2)^(3/2))
Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.53, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 4140, 3042, 4613, 3042, 3070, 244, 2009}
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.
\(\displaystyle \int \frac {\cot ^2(x)}{\left (a \tan ^2(x)+a\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (x)^2 \left (a \tan (x)^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \frac {\cot ^2(x)}{\left (a \sec ^2(x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (x)^2 \left (a \sec (x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4613 |
\(\displaystyle \frac {\sec (x) \int \cos ^3(x) \cot ^2(x)dx}{a \sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (x) \int \sin \left (x+\frac {\pi }{2}\right )^3 \tan \left (x+\frac {\pi }{2}\right )^2dx}{a \sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 3070 |
\(\displaystyle -\frac {\sec (x) \int \csc ^2(x) \left (1-\sin ^2(x)\right )^2d(-\sin (x))}{a \sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {\sec (x) \int \left (\csc ^2(x)+\sin ^2(x)-2\right )d(-\sin (x))}{a \sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sec (x) \left (-\frac {1}{3} \sin ^3(x)+2 \sin (x)+\csc (x)\right )}{a \sqrt {a \sec ^2(x)}}\) |
Input:
Int[Cot[x]^2/(a + a*Tan[x]^2)^(3/2),x]
Output:
-((Sec[x]*(Csc[x] + 2*Sin[x] - Sin[x]^3/3))/(a*Sqrt[a*Sec[x]^2]))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[(u_.)*((b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sec[e + f*x]^ n)^FracPart[p]/(Sec[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Se c[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 1.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {\frac {\cos \left (x \right )^{2} \cot \left (x \right )}{3}+\frac {4 \cot \left (x \right )}{3}-\frac {8 \sec \left (x \right ) \csc \left (x \right )}{3}}{\sqrt {a \sec \left (x \right )^{2}}\, a}\) | \(32\) |
risch | \(\frac {i \left ({\mathrm e}^{6 i x}+20+20 \,{\mathrm e}^{4 i x}-89 \cos \left (2 x \right )-91 i \sin \left (2 x \right )\right )}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}\) | \(70\) |
Input:
int(cot(x)^2/(a+a*tan(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
(1/3*cos(x)^2*cot(x)+4/3*cot(x)-8/3*sec(x)*csc(x))/(a*sec(x)^2)^(1/2)/a
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {{\left (8 \, \tan \left (x\right )^{4} + 12 \, \tan \left (x\right )^{2} + 3\right )} \sqrt {a \tan \left (x\right )^{2} + a}}{3 \, {\left (a^{2} \tan \left (x\right )^{5} + 2 \, a^{2} \tan \left (x\right )^{3} + a^{2} \tan \left (x\right )\right )}} \] Input:
integrate(cot(x)^2/(a+a*tan(x)^2)^(3/2),x, algorithm="fricas")
Output:
-1/3*(8*tan(x)^4 + 12*tan(x)^2 + 3)*sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^5 + 2 *a^2*tan(x)^3 + a^2*tan(x))
\[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(x)**2/(a+a*tan(x)**2)**(3/2),x)
Output:
Integral(cot(x)**2/(a*(tan(x)**2 + 1))**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (52) = 104\).
Time = 0.16 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.75 \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {{\left ({\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (8 \, x\right ) + 20 \, {\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (6 \, x\right ) + 10 \, {\left (9 \, \sin \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) - {\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (8 \, x\right ) - 20 \, {\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (6 \, x\right ) - {\left (90 \, \cos \left (4 \, x\right ) - 20 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) - 90 \, \cos \left (3 \, x\right ) \sin \left (4 \, x\right ) - {\left (20 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) + 90 \, \cos \left (4 \, x\right ) \sin \left (3 \, x\right ) + 20 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right )\right )} \sqrt {a}}{24 \, {\left (a^{2} \cos \left (5 \, x\right )^{2} - 2 \, a^{2} \cos \left (5 \, x\right ) \cos \left (3 \, x\right ) + a^{2} \cos \left (3 \, x\right )^{2} + a^{2} \sin \left (5 \, x\right )^{2} - 2 \, a^{2} \sin \left (5 \, x\right ) \sin \left (3 \, x\right ) + a^{2} \sin \left (3 \, x\right )^{2}\right )}} \] Input:
integrate(cot(x)^2/(a+a*tan(x)^2)^(3/2),x, algorithm="maxima")
Output:
1/24*((sin(5*x) - sin(3*x))*cos(8*x) + 20*(sin(5*x) - sin(3*x))*cos(6*x) + 10*(9*sin(4*x) - 2*sin(2*x))*cos(5*x) - (cos(5*x) - cos(3*x))*sin(8*x) - 20*(cos(5*x) - cos(3*x))*sin(6*x) - (90*cos(4*x) - 20*cos(2*x) - 1)*sin(5* x) - 90*cos(3*x)*sin(4*x) - (20*cos(2*x) + 1)*sin(3*x) + 90*cos(4*x)*sin(3 *x) + 20*cos(3*x)*sin(2*x))*sqrt(a)/(a^2*cos(5*x)^2 - 2*a^2*cos(5*x)*cos(3 *x) + a^2*cos(3*x)^2 + a^2*sin(5*x)^2 - 2*a^2*sin(5*x)*sin(3*x) + a^2*sin( 3*x)^2)
Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {{\left (5 \, \tan \left (x\right )^{2} + 6\right )} \tan \left (x\right )}{3 \, {\left (a \tan \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} + \frac {2}{{\left ({\left (\sqrt {a} \tan \left (x\right ) - \sqrt {a \tan \left (x\right )^{2} + a}\right )}^{2} - a\right )} \sqrt {a}} \] Input:
integrate(cot(x)^2/(a+a*tan(x)^2)^(3/2),x, algorithm="giac")
Output:
-1/3*(5*tan(x)^2 + 6)*tan(x)/(a*tan(x)^2 + a)^(3/2) + 2/(((sqrt(a)*tan(x) - sqrt(a*tan(x)^2 + a))^2 - a)*sqrt(a))
Timed out. \[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (x\right )}^2}{{\left (a\,{\mathrm {tan}\left (x\right )}^2+a\right )}^{3/2}} \,d x \] Input:
int(cot(x)^2/(a + a*tan(x)^2)^(3/2),x)
Output:
int(cot(x)^2/(a + a*tan(x)^2)^(3/2), x)
\[ \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\tan \left (x \right )^{2}+1}\, \cot \left (x \right )^{2}}{\tan \left (x \right )^{4}+2 \tan \left (x \right )^{2}+1}d x \right )}{a^{2}} \] Input:
int(cot(x)^2/(a+a*tan(x)^2)^(3/2),x)
Output:
(sqrt(a)*int((sqrt(tan(x)**2 + 1)*cot(x)**2)/(tan(x)**4 + 2*tan(x)**2 + 1) ,x))/a**2