Integrand size = 21, antiderivative size = 47 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=-\frac {4}{9} \cot (9 x)-\frac {1}{27} \cot ^3(9 x)+\frac {2}{3} \tan (9 x)+\frac {4}{27} \tan ^3(9 x)+\frac {1}{45} \tan ^5(9 x) \] Output:
-4/9*cot(9*x)-1/27*cot(9*x)^3+2/3*tan(9*x)+4/27*tan(9*x)^3+1/45*tan(9*x)^5
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=-\frac {11}{27} \cot (9 x)-\frac {1}{27} \cot (9 x) \csc ^2(9 x)+\frac {73}{135} \tan (9 x)+\frac {14}{135} \sec ^2(9 x) \tan (9 x)+\frac {1}{45} \sec ^4(9 x) \tan (9 x) \] Input:
Integrate[(1 + Cot[9*x]^2)^2*(1 + Tan[9*x]^2)^3,x]
Output:
(-11*Cot[9*x])/27 - (Cot[9*x]*Csc[9*x]^2)/27 + (73*Tan[9*x])/135 + (14*Sec [9*x]^2*Tan[9*x])/135 + (Sec[9*x]^4*Tan[9*x])/45
Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4140, 3042, 4140, 3042, 3100, 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.
\(\displaystyle \int \left (\tan ^2(9 x)+1\right )^3 \left (\cot ^2(9 x)+1\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\tan (9 x)^2+1\right )^3 \left (\cot (9 x)^2+1\right )^2dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \left (\cot ^2(9 x)+1\right )^2 \sec ^6(9 x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (\tan \left (9 x+\frac {\pi }{2}\right )^2+1\right )^2}{\sin \left (9 x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \csc ^4(9 x) \sec ^6(9 x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (9 x)^4 \sec (9 x)^6dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {1}{9} \int \cot ^4(9 x) \left (\tan ^2(9 x)+1\right )^4d\tan (9 x)\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {1}{9} \int \left (\cot ^4(9 x)+4 \cot ^2(9 x)+\tan ^4(9 x)+4 \tan ^2(9 x)+6\right )d\tan (9 x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{5} \tan ^5(9 x)+\frac {4}{3} \tan ^3(9 x)+6 \tan (9 x)-\frac {1}{3} \cot ^3(9 x)-4 \cot (9 x)\right )\) |
Input:
Int[(1 + Cot[9*x]^2)^2*(1 + Tan[9*x]^2)^3,x]
Output:
(-4*Cot[9*x] - Cot[9*x]^3/3 + 6*Tan[9*x] + (4*Tan[9*x]^3)/3 + Tan[9*x]^5/5 )/9
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[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/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]
Time = 0.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {4 \cot \left (9 x \right )}{9}-\frac {\cot \left (9 x \right )^{3}}{27}+\frac {2 \tan \left (9 x \right )}{3}+\frac {4 \tan \left (9 x \right )^{3}}{27}+\frac {\tan \left (9 x \right )^{5}}{45}\) | \(38\) |
default | \(-\frac {4 \cot \left (9 x \right )}{9}-\frac {\cot \left (9 x \right )^{3}}{27}+\frac {2 \tan \left (9 x \right )}{3}+\frac {4 \tan \left (9 x \right )^{3}}{27}+\frac {\tan \left (9 x \right )^{5}}{45}\) | \(38\) |
parallelrisch | \(-\frac {4 \cot \left (9 x \right )}{9}-\frac {\cot \left (9 x \right )^{3}}{27}+\frac {2 \tan \left (9 x \right )}{3}+\frac {4 \tan \left (9 x \right )^{3}}{27}+\frac {\tan \left (9 x \right )^{5}}{45}\) | \(38\) |
norman | \(\frac {-\frac {1}{27}-\frac {4 \tan \left (9 x \right )^{2}}{9}+\frac {2 \tan \left (9 x \right )^{4}}{3}+\frac {4 \tan \left (9 x \right )^{6}}{27}+\frac {\tan \left (9 x \right )^{8}}{45}}{\tan \left (9 x \right )^{3}}\) | \(42\) |
risch | \(\frac {256 i \left (6 \,{\mathrm e}^{54 i x}+2 \,{\mathrm e}^{36 i x}-2 \,{\mathrm e}^{18 i x}-1\right )}{135 \left ({\mathrm e}^{18 i x}+1\right )^{5} \left ({\mathrm e}^{18 i x}-1\right )^{3}}\) | \(45\) |
parts | \(x +\frac {\tan \left (9 x \right )^{5}}{45}+\frac {4 \tan \left (9 x \right )^{3}}{27}+\frac {2 \tan \left (9 x \right )}{3}-\frac {\arctan \left (\tan \left (9 x \right )\right )}{9}-\frac {4 \cot \left (9 x \right )}{9}-\frac {\cot \left (9 x \right )^{3}}{27}\) | \(46\) |
Input:
int((1+cot(9*x)^2)^2*(1+tan(9*x)^2)^3,x,method=_RETURNVERBOSE)
Output:
-4/9*cot(9*x)-1/27*cot(9*x)^3+2/3*tan(9*x)+4/27*tan(9*x)^3+1/45*tan(9*x)^5
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {3 \, \tan \left (9 \, x\right )^{8} + 20 \, \tan \left (9 \, x\right )^{6} + 90 \, \tan \left (9 \, x\right )^{4} - 60 \, \tan \left (9 \, x\right )^{2} - 5}{135 \, \tan \left (9 \, x\right )^{3}} \] Input:
integrate((1+cot(9*x)^2)^2*(1+tan(9*x)^2)^3,x, algorithm="fricas")
Output:
1/135*(3*tan(9*x)^8 + 20*tan(9*x)^6 + 90*tan(9*x)^4 - 60*tan(9*x)^2 - 5)/t an(9*x)^3
Time = 2.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {\tan ^{5}{\left (9 x \right )}}{45} + \frac {4 \tan ^{3}{\left (9 x \right )}}{27} + \frac {2 \tan {\left (9 x \right )}}{3} - \frac {4}{9 \tan {\left (9 x \right )}} - \frac {1}{27 \tan ^{3}{\left (9 x \right )}} \] Input:
integrate((1+cot(9*x)**2)**2*(1+tan(9*x)**2)**3,x)
Output:
tan(9*x)**5/45 + 4*tan(9*x)**3/27 + 2*tan(9*x)/3 - 4/(9*tan(9*x)) - 1/(27* tan(9*x)**3)
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {1}{45} \, \tan \left (9 \, x\right )^{5} + \frac {4}{27} \, \tan \left (9 \, x\right )^{3} - \frac {12 \, \tan \left (9 \, x\right )^{2} + 1}{27 \, \tan \left (9 \, x\right )^{3}} + \frac {2}{3} \, \tan \left (9 \, x\right ) \] Input:
integrate((1+cot(9*x)^2)^2*(1+tan(9*x)^2)^3,x, algorithm="maxima")
Output:
1/45*tan(9*x)^5 + 4/27*tan(9*x)^3 - 1/27*(12*tan(9*x)^2 + 1)/tan(9*x)^3 + 2/3*tan(9*x)
Time = 69.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {1}{45} \, \tan \left (9 \, x\right )^{5} + \frac {4}{27} \, \tan \left (9 \, x\right )^{3} - \frac {12 \, \tan \left (9 \, x\right )^{2} + 1}{27 \, \tan \left (9 \, x\right )^{3}} + \frac {2}{3} \, \tan \left (9 \, x\right ) \] Input:
integrate((1+cot(9*x)^2)^2*(1+tan(9*x)^2)^3,x, algorithm="giac")
Output:
1/45*tan(9*x)^5 + 4/27*tan(9*x)^3 - 1/27*(12*tan(9*x)^2 + 1)/tan(9*x)^3 + 2/3*tan(9*x)
Time = 18.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {3\,{\mathrm {tan}\left (9\,x\right )}^8+20\,{\mathrm {tan}\left (9\,x\right )}^6+90\,{\mathrm {tan}\left (9\,x\right )}^4-60\,{\mathrm {tan}\left (9\,x\right )}^2-5}{135\,{\mathrm {tan}\left (9\,x\right )}^3} \] Input:
int((tan(9*x)^2 + 1)^3*(cot(9*x)^2 + 1)^2,x)
Output:
(90*tan(9*x)^4 - 60*tan(9*x)^2 + 20*tan(9*x)^6 + 3*tan(9*x)^8 - 5)/(135*ta n(9*x)^3)
Time = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 3.28 \[ \int \left (1+\cot ^2(9 x)\right )^2 \left (1+\tan ^2(9 x)\right )^3 \, dx=\frac {3 \cos \left (9 x \right ) \sin \left (9 x \right )^{5} \tan \left (9 x \right )^{5}+10 \cos \left (9 x \right ) \sin \left (9 x \right )^{5} \tan \left (9 x \right )^{3}+15 \cos \left (9 x \right ) \sin \left (9 x \right )^{5} \tan \left (9 x \right )-3 \cos \left (9 x \right ) \sin \left (9 x \right )^{3} \tan \left (9 x \right )^{5}-10 \cos \left (9 x \right ) \sin \left (9 x \right )^{3} \tan \left (9 x \right )^{3}-15 \cos \left (9 x \right ) \sin \left (9 x \right )^{3} \tan \left (9 x \right )+120 \sin \left (9 x \right )^{6}-180 \sin \left (9 x \right )^{4}+45 \sin \left (9 x \right )^{2}+5}{135 \cos \left (9 x \right ) \sin \left (9 x \right )^{3} \left (\sin \left (9 x \right )^{2}-1\right )} \] Input:
int((1+cot(9*x)^2)^2*(1+tan(9*x)^2)^3,x)
Output:
(3*cos(9*x)*sin(9*x)**5*tan(9*x)**5 + 10*cos(9*x)*sin(9*x)**5*tan(9*x)**3 + 15*cos(9*x)*sin(9*x)**5*tan(9*x) - 3*cos(9*x)*sin(9*x)**3*tan(9*x)**5 - 10*cos(9*x)*sin(9*x)**3*tan(9*x)**3 - 15*cos(9*x)*sin(9*x)**3*tan(9*x) + 1 20*sin(9*x)**6 - 180*sin(9*x)**4 + 45*sin(9*x)**2 + 5)/(135*cos(9*x)*sin(9 *x)**3*(sin(9*x)**2 - 1))