Integrand size = 23, antiderivative size = 297 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=-\frac {x}{(a-b)^3}+\frac {b^{7/2} \left (99 a^2-154 a b+63 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{11/2} (a-b)^3 f}-\frac {\left (8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right ) \cot (e+f x)}{8 a^5 (a-b)^2 f}+\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{24 a^4 (a-b)^2 f}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{40 a^3 (a-b)^2 f}-\frac {b \cot ^5(e+f x)}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {(13 a-9 b) b \cot ^5(e+f x)}{8 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )} \] Output:
-x/(a-b)^3+1/8*b^(7/2)*(99*a^2-154*a*b+63*b^2)*arctan(b^(1/2)*tan(f*x+e)/a ^(1/2))/a^(11/2)/(a-b)^3/f-1/8*(8*a^4+8*a^3*b+8*a^2*b^2-91*a*b^3+63*b^4)*c ot(f*x+e)/a^5/(a-b)^2/f+1/24*(8*a^3+8*a^2*b-91*a*b^2+63*b^3)*cot(f*x+e)^3/ a^4/(a-b)^2/f-1/40*(8*a^2-91*a*b+63*b^2)*cot(f*x+e)^5/a^3/(a-b)^2/f-1/4*b* cot(f*x+e)^5/a/(a-b)/f/(a+b*tan(f*x+e)^2)^2-1/8*(13*a-9*b)*b*cot(f*x+e)^5/ a^2/(a-b)^2/f/(a+b*tan(f*x+e)^2)
Leaf count is larger than twice the leaf count of optimal. \(949\) vs. \(2(297)=594\).
Time = 6.22 (sec) , antiderivative size = 949, normalized size of antiderivative = 3.20 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {b^{7/2} \left (99 a^2-154 a b+63 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{11/2} (a-b)^3 f}+\frac {\csc ^5(e+f x) \left (-3184 a^7 \cos (e+f x)+7440 a^6 b \cos (e+f x)-12000 a^5 b^2 \cos (e+f x)+10240 a^4 b^3 \cos (e+f x)+6450 a^3 b^4 \cos (e+f x)+714 a^2 b^5 \cos (e+f x)-22890 a b^6 \cos (e+f x)+13230 b^7 \cos (e+f x)-1536 a^7 \cos (3 (e+f x))+7648 a^6 b \cos (3 (e+f x))-2912 a^5 b^2 \cos (3 (e+f x))-1152 a^4 b^3 \cos (3 (e+f x))-14872 a^3 b^4 \cos (3 (e+f x))-12796 a^2 b^5 \cos (3 (e+f x))+52080 a b^6 \cos (3 (e+f x))-26460 b^7 \cos (3 (e+f x))-704 a^7 \cos (5 (e+f x))+2656 a^6 b \cos (5 (e+f x))-4128 a^5 b^2 \cos (5 (e+f x))-3712 a^4 b^3 \cos (5 (e+f x))+5504 a^3 b^4 \cos (5 (e+f x))+27684 a^2 b^5 \cos (5 (e+f x))-46200 a b^6 \cos (5 (e+f x))+18900 b^7 \cos (5 (e+f x))-536 a^7 \cos (7 (e+f x))+248 a^6 b \cos (7 (e+f x))+768 a^5 b^2 \cos (7 (e+f x))+128 a^4 b^3 \cos (7 (e+f x))+6553 a^3 b^4 \cos (7 (e+f x))-21441 a^2 b^5 \cos (7 (e+f x))+20895 a b^6 \cos (7 (e+f x))-6615 b^7 \cos (7 (e+f x))-184 a^7 \cos (9 (e+f x))+440 a^6 b \cos (9 (e+f x))-160 a^5 b^2 \cos (9 (e+f x))+640 a^4 b^3 \cos (9 (e+f x))-3635 a^3 b^4 \cos (9 (e+f x))+5839 a^2 b^5 \cos (9 (e+f x))-3885 a b^6 \cos (9 (e+f x))+945 b^7 \cos (9 (e+f x))-720 a^7 (e+f x) \sin (e+f x)-3360 a^6 b (e+f x) \sin (e+f x)-15120 a^5 b^2 (e+f x) \sin (e+f x)-480 a^7 (e+f x) \sin (3 (e+f x))+10080 a^5 b^2 (e+f x) \sin (3 (e+f x))+480 a^7 (e+f x) \sin (5 (e+f x))+1920 a^6 b (e+f x) \sin (5 (e+f x))-4320 a^5 b^2 (e+f x) \sin (5 (e+f x))+120 a^7 (e+f x) \sin (7 (e+f x))-1200 a^6 b (e+f x) \sin (7 (e+f x))+1080 a^5 b^2 (e+f x) \sin (7 (e+f x))-120 a^7 (e+f x) \sin (9 (e+f x))+240 a^6 b (e+f x) \sin (9 (e+f x))-120 a^5 b^2 (e+f x) \sin (9 (e+f x))\right )}{7680 a^5 (a-b)^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2} \] Input:
Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]
Output:
(b^(7/2)*(99*a^2 - 154*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a] ])/(8*a^(11/2)*(a - b)^3*f) + (Csc[e + f*x]^5*(-3184*a^7*Cos[e + f*x] + 74 40*a^6*b*Cos[e + f*x] - 12000*a^5*b^2*Cos[e + f*x] + 10240*a^4*b^3*Cos[e + f*x] + 6450*a^3*b^4*Cos[e + f*x] + 714*a^2*b^5*Cos[e + f*x] - 22890*a*b^6 *Cos[e + f*x] + 13230*b^7*Cos[e + f*x] - 1536*a^7*Cos[3*(e + f*x)] + 7648* a^6*b*Cos[3*(e + f*x)] - 2912*a^5*b^2*Cos[3*(e + f*x)] - 1152*a^4*b^3*Cos[ 3*(e + f*x)] - 14872*a^3*b^4*Cos[3*(e + f*x)] - 12796*a^2*b^5*Cos[3*(e + f *x)] + 52080*a*b^6*Cos[3*(e + f*x)] - 26460*b^7*Cos[3*(e + f*x)] - 704*a^7 *Cos[5*(e + f*x)] + 2656*a^6*b*Cos[5*(e + f*x)] - 4128*a^5*b^2*Cos[5*(e + f*x)] - 3712*a^4*b^3*Cos[5*(e + f*x)] + 5504*a^3*b^4*Cos[5*(e + f*x)] + 27 684*a^2*b^5*Cos[5*(e + f*x)] - 46200*a*b^6*Cos[5*(e + f*x)] + 18900*b^7*Co s[5*(e + f*x)] - 536*a^7*Cos[7*(e + f*x)] + 248*a^6*b*Cos[7*(e + f*x)] + 7 68*a^5*b^2*Cos[7*(e + f*x)] + 128*a^4*b^3*Cos[7*(e + f*x)] + 6553*a^3*b^4* Cos[7*(e + f*x)] - 21441*a^2*b^5*Cos[7*(e + f*x)] + 20895*a*b^6*Cos[7*(e + f*x)] - 6615*b^7*Cos[7*(e + f*x)] - 184*a^7*Cos[9*(e + f*x)] + 440*a^6*b* Cos[9*(e + f*x)] - 160*a^5*b^2*Cos[9*(e + f*x)] + 640*a^4*b^3*Cos[9*(e + f *x)] - 3635*a^3*b^4*Cos[9*(e + f*x)] + 5839*a^2*b^5*Cos[9*(e + f*x)] - 388 5*a*b^6*Cos[9*(e + f*x)] + 945*b^7*Cos[9*(e + f*x)] - 720*a^7*(e + f*x)*Si n[e + f*x] - 3360*a^6*b*(e + f*x)*Sin[e + f*x] - 15120*a^5*b^2*(e + f*x)*S in[e + f*x] - 480*a^7*(e + f*x)*Sin[3*(e + f*x)] + 10080*a^5*b^2*(e + f...
Time = 1.01 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4153, 374, 441, 445, 27, 445, 27, 445, 397, 216, 218}
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.
| \(\displaystyle \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^6 \left (a+b \tan (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^3}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^6(e+f x) \left (-9 b \tan ^2(e+f x)+4 a-9 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cot ^6(e+f x) \left (8 a^2-91 b a+63 b^2-7 (13 a-9 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {5 \cot ^4(e+f x) \left (8 a^3+8 b a^2-91 b^2 a+63 b^3+b \left (8 a^2-91 b a+63 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{5 a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\cot ^4(e+f x) \left (8 a^3+8 b a^2-91 b^2 a+63 b^3+b \left (8 a^2-91 b a+63 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {3 \cot ^2(e+f x) \left (8 a^4+8 b a^3+8 b^2 a^2-91 b^3 a+63 b^4+b \left (8 a^3+8 b a^2-91 b^2 a+63 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{3 a}-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {\cot ^2(e+f x) \left (8 a^4+8 b a^3+8 b^2 a^2-91 b^3 a+63 b^4+b \left (8 a^3+8 b a^2-91 b^2 a+63 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\int \frac {8 a^5+8 b a^4+8 b^2 a^3+8 b^3 a^2-91 b^4 a+63 b^5+b \left (8 a^4+8 b a^3+8 b^2 a^2-91 b^3 a+63 b^4\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {\left (8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right ) \cot (e+f x)}{a}}{a}-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\frac {8 a^5 \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a-b}-\frac {b^4 \left (99 a^2-154 a b+63 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}}{a}-\frac {\left (8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right ) \cot (e+f x)}{a}}{a}-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\frac {8 a^5 \arctan (\tan (e+f x))}{a-b}-\frac {b^4 \left (99 a^2-154 a b+63 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}}{a}-\frac {\left (8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right ) \cot (e+f x)}{a}}{a}-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\left (8 a^2-91 a b+63 b^2\right ) \cot ^5(e+f x)}{5 a}-\frac {-\frac {\left (8 a^3+8 a^2 b-91 a b^2+63 b^3\right ) \cot ^3(e+f x)}{3 a}-\frac {-\frac {\frac {8 a^5 \arctan (\tan (e+f x))}{a-b}-\frac {b^{7/2} \left (99 a^2-154 a b+63 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)}}{a}-\frac {\left (8 a^4+8 a^3 b+8 a^2 b^2-91 a b^3+63 b^4\right ) \cot (e+f x)}{a}}{a}}{a}}{2 a (a-b)}-\frac {b (13 a-9 b) \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{4 a (a-b)}-\frac {b \cot ^5(e+f x)}{4 a (a-b) \left (a+b \tan ^2(e+f x)\right )^2}}{f}\) |
Input:
Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]
Output:
(-1/4*(b*Cot[e + f*x]^5)/(a*(a - b)*(a + b*Tan[e + f*x]^2)^2) + ((-1/5*((8 *a^2 - 91*a*b + 63*b^2)*Cot[e + f*x]^5)/a - (-1/3*((8*a^3 + 8*a^2*b - 91*a *b^2 + 63*b^3)*Cot[e + f*x]^3)/a - (-(((8*a^5*ArcTan[Tan[e + f*x]])/(a - b ) - (b^(7/2)*(99*a^2 - 154*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqr t[a]])/(Sqrt[a]*(a - b)))/a) - ((8*a^4 + 8*a^3*b + 8*a^2*b^2 - 91*a*b^3 + 63*b^4)*Cot[e + f*x])/a)/a)/a)/(2*a*(a - b)) - ((13*a - 9*b)*b*Cot[e + f*x ]^5)/(2*a*(a - b)*(a + b*Tan[e + f*x]^2)))/(4*a*(a - b)))/f
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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 = 0.86 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {b^{4} \left (\frac {\left (\frac {19}{8} a^{2} b -\frac {17}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (21 a^{2}-38 a b +17 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (99 a^{2}-154 a b +63 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5} \left (a -b \right )^{3}}-\frac {1}{5 a^{3} \tan \left (f x +e \right )^{5}}-\frac {-3 b -a}{3 a^{4} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+3 a b +6 b^{2}}{a^{5} \tan \left (f x +e \right )}}{f}\) | \(199\) |
| default | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {b^{4} \left (\frac {\left (\frac {19}{8} a^{2} b -\frac {17}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (21 a^{2}-38 a b +17 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (99 a^{2}-154 a b +63 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5} \left (a -b \right )^{3}}-\frac {1}{5 a^{3} \tan \left (f x +e \right )^{5}}-\frac {-3 b -a}{3 a^{4} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+3 a b +6 b^{2}}{a^{5} \tan \left (f x +e \right )}}{f}\) | \(199\) |
| risch | \(\text {Expression too large to display}\) | \(1483\) |
Input:
int(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/f*(-1/(a-b)^3*arctan(tan(f*x+e))+b^4/a^5/(a-b)^3*(((19/8*a^2*b-17/4*a*b^ 2+15/8*b^3)*tan(f*x+e)^3+1/8*a*(21*a^2-38*a*b+17*b^2)*tan(f*x+e))/(a+b*tan (f*x+e)^2)^2+1/8*(99*a^2-154*a*b+63*b^2)/(a*b)^(1/2)*arctan(b*tan(f*x+e)/( a*b)^(1/2)))-1/5/a^3/tan(f*x+e)^5-1/3*(-3*b-a)/a^4/tan(f*x+e)^3-(a^2+3*a*b +6*b^2)/a^5/tan(f*x+e))
Time = 0.19 (sec) , antiderivative size = 1114, normalized size of antiderivative = 3.75 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="fricas")
Output:
[-1/480*(480*a^5*b^2*f*x*tan(f*x + e)^9 + 960*a^6*b*f*x*tan(f*x + e)^7 + 4 80*a^7*f*x*tan(f*x + e)^5 + 60*(8*a^5*b^2 - 99*a^2*b^5 + 154*a*b^6 - 63*b^ 7)*tan(f*x + e)^8 + 96*a^7 - 288*a^6*b + 288*a^5*b^2 - 96*a^4*b^3 + 20*(48 *a^6*b - 8*a^5*b^2 - 495*a^3*b^4 + 770*a^2*b^5 - 315*a*b^6)*tan(f*x + e)^6 + 32*(15*a^7 - 10*a^6*b + 3*a^5*b^2 - 99*a^4*b^3 + 154*a^3*b^4 - 63*a^2*b ^5)*tan(f*x + e)^4 - 32*(5*a^7 - 6*a^6*b - 12*a^5*b^2 + 22*a^4*b^3 - 9*a^3 *b^4)*tan(f*x + e)^2 + 15*((99*a^2*b^5 - 154*a*b^6 + 63*b^7)*tan(f*x + e)^ 9 + 2*(99*a^3*b^4 - 154*a^2*b^5 + 63*a*b^6)*tan(f*x + e)^7 + (99*a^4*b^3 - 154*a^3*b^4 + 63*a^2*b^5)*tan(f*x + e)^5)*sqrt(-b/a)*log((b^2*tan(f*x + e )^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e ))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^8*b ^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8* b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^5), -1/240*(240*a^5*b^2*f*x*tan(f*x + e)^9 + 480 *a^6*b*f*x*tan(f*x + e)^7 + 240*a^7*f*x*tan(f*x + e)^5 + 30*(8*a^5*b^2 - 9 9*a^2*b^5 + 154*a*b^6 - 63*b^7)*tan(f*x + e)^8 + 48*a^7 - 144*a^6*b + 144* a^5*b^2 - 48*a^4*b^3 + 10*(48*a^6*b - 8*a^5*b^2 - 495*a^3*b^4 + 770*a^2*b^ 5 - 315*a*b^6)*tan(f*x + e)^6 + 16*(15*a^7 - 10*a^6*b + 3*a^5*b^2 - 99*a^4 *b^3 + 154*a^3*b^4 - 63*a^2*b^5)*tan(f*x + e)^4 - 16*(5*a^7 - 6*a^6*b - 12 *a^5*b^2 + 22*a^4*b^3 - 9*a^3*b^4)*tan(f*x + e)^2 - 15*((99*a^2*b^5 - 1...
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(cot(f*x+e)**6/(a+b*tan(f*x+e)**2)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.33 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {\frac {15 \, {\left (99 \, a^{2} b^{4} - 154 \, a b^{5} + 63 \, b^{6}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right )} \sqrt {a b}} - \frac {15 \, {\left (8 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 8 \, a^{2} b^{4} - 91 \, a b^{5} + 63 \, b^{6}\right )} \tan \left (f x + e\right )^{8} + 5 \, {\left (48 \, a^{5} b + 40 \, a^{4} b^{2} + 40 \, a^{3} b^{3} - 455 \, a^{2} b^{4} + 315 \, a b^{5}\right )} \tan \left (f x + e\right )^{6} + 24 \, a^{6} - 48 \, a^{5} b + 24 \, a^{4} b^{2} + 8 \, {\left (15 \, a^{6} + 5 \, a^{5} b + 8 \, a^{4} b^{2} - 91 \, a^{3} b^{3} + 63 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{4} - 8 \, {\left (5 \, a^{6} - a^{5} b - 13 \, a^{4} b^{2} + 9 \, a^{3} b^{3}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{7} b^{2} - 2 \, a^{6} b^{3} + a^{5} b^{4}\right )} \tan \left (f x + e\right )^{9} + 2 \, {\left (a^{8} b - 2 \, a^{7} b^{2} + a^{6} b^{3}\right )} \tan \left (f x + e\right )^{7} + {\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} \tan \left (f x + e\right )^{5}} - \frac {120 \, {\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}}}{120 \, f} \] Input:
integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")
Output:
1/120*(15*(99*a^2*b^4 - 154*a*b^5 + 63*b^6)*arctan(b*tan(f*x + e)/sqrt(a*b ))/((a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*sqrt(a*b)) - (15*(8*a^4*b^2 + 8* a^3*b^3 + 8*a^2*b^4 - 91*a*b^5 + 63*b^6)*tan(f*x + e)^8 + 5*(48*a^5*b + 40 *a^4*b^2 + 40*a^3*b^3 - 455*a^2*b^4 + 315*a*b^5)*tan(f*x + e)^6 + 24*a^6 - 48*a^5*b + 24*a^4*b^2 + 8*(15*a^6 + 5*a^5*b + 8*a^4*b^2 - 91*a^3*b^3 + 63 *a^2*b^4)*tan(f*x + e)^4 - 8*(5*a^6 - a^5*b - 13*a^4*b^2 + 9*a^3*b^3)*tan( f*x + e)^2)/((a^7*b^2 - 2*a^6*b^3 + a^5*b^4)*tan(f*x + e)^9 + 2*(a^8*b - 2 *a^7*b^2 + a^6*b^3)*tan(f*x + e)^7 + (a^9 - 2*a^8*b + a^7*b^2)*tan(f*x + e )^5) - 120*(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3))/f
Time = 0.65 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\frac {{\left (99 \, a^{2} b^{4} - 154 \, a b^{5} + 63 \, b^{6}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{8 \, {\left (a^{8} f - 3 \, a^{7} b f + 3 \, a^{6} b^{2} f - a^{5} b^{3} f\right )} \sqrt {a b}} - \frac {f x + e}{a^{3} f - 3 \, a^{2} b f + 3 \, a b^{2} f - b^{3} f} + \frac {19 \, a b^{5} \tan \left (f x + e\right )^{3} - 15 \, b^{6} \tan \left (f x + e\right )^{3} + 21 \, a^{2} b^{4} \tan \left (f x + e\right ) - 17 \, a b^{5} \tan \left (f x + e\right )}{8 \, {\left (a^{7} f - 2 \, a^{6} b f + a^{5} b^{2} f\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}} - \frac {15 \, a^{2} \tan \left (f x + e\right )^{4} + 45 \, a b \tan \left (f x + e\right )^{4} + 90 \, b^{2} \tan \left (f x + e\right )^{4} - 5 \, a^{2} \tan \left (f x + e\right )^{2} - 15 \, a b \tan \left (f x + e\right )^{2} + 3 \, a^{2}}{15 \, a^{5} f \tan \left (f x + e\right )^{5}} \] Input:
integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")
Output:
1/8*(99*a^2*b^4 - 154*a*b^5 + 63*b^6)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a ^8*f - 3*a^7*b*f + 3*a^6*b^2*f - a^5*b^3*f)*sqrt(a*b)) - (f*x + e)/(a^3*f - 3*a^2*b*f + 3*a*b^2*f - b^3*f) + 1/8*(19*a*b^5*tan(f*x + e)^3 - 15*b^6*t an(f*x + e)^3 + 21*a^2*b^4*tan(f*x + e) - 17*a*b^5*tan(f*x + e))/((a^7*f - 2*a^6*b*f + a^5*b^2*f)*(b*tan(f*x + e)^2 + a)^2) - 1/15*(15*a^2*tan(f*x + e)^4 + 45*a*b*tan(f*x + e)^4 + 90*b^2*tan(f*x + e)^4 - 5*a^2*tan(f*x + e) ^2 - 15*a*b*tan(f*x + e)^2 + 3*a^2)/(a^5*f*tan(f*x + e)^5)
Time = 12.46 (sec) , antiderivative size = 2507, normalized size of antiderivative = 8.44 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(cot(e + f*x)^6/(a + b*tan(e + f*x)^2)^3,x)
Output:
(atan((b^5*tan(e + f*x)*(-a^11*b^7)^(3/2)*3969i - a*b^4*tan(e + f*x)*(-a^1 1*b^7)^(3/2)*19404i + a^4*b*tan(e + f*x)*(-a^11*b^7)^(3/2)*9801i + a^22*b* tan(e + f*x)*(-a^11*b^7)^(1/2)*64i + a^2*b^3*tan(e + f*x)*(-a^11*b^7)^(3/2 )*36190i - a^3*b^2*tan(e + f*x)*(-a^11*b^7)^(3/2)*30492i)/(3969*a^17*b^15 - 19404*a^18*b^14 + 36190*a^19*b^13 - 30492*a^20*b^12 + 9801*a^21*b^11 - 6 4*a^28*b^4))*(-a^11*b^7)^(1/2)*(99*a^2 - 154*a*b + 63*b^2)*1i)/(8*f*(3*a^1 3*b - a^14 + a^11*b^3 - 3*a^12*b^2)) - (1/(5*a) + (tan(e + f*x)^4*(35*a*b + 15*a^2 + 63*b^2))/(15*a^3) - (tan(e + f*x)^2*(5*a + 9*b))/(15*a^2) + (ta n(e + f*x)^6*(48*a^4*b - 455*a*b^4 + 315*b^5 + 40*a^2*b^3 + 40*a^3*b^2))/( 24*a^4*(a^2 - 2*a*b + b^2)) + (tan(e + f*x)^8*(63*b^6 - 91*a*b^5 + 8*a^2*b ^4 + 8*a^3*b^3 + 8*a^4*b^2))/(8*a^5*(a^2 - 2*a*b + b^2)))/(f*(a^2*tan(e + f*x)^5 + b^2*tan(e + f*x)^9 + 2*a*b*tan(e + f*x)^7)) - (2*atan(((((1032192 *a^20*b^17 - 11812864*a^21*b^16 + 61489152*a^22*b^15 - 192135168*a^23*b^14 + 400392192*a^24*b^13 - 584220672*a^25*b^12 + 608862208*a^26*b^11 - 45229 6704*a^27*b^10 + 231653376*a^28*b^9 - 71122944*a^29*b^8 + 606208*a^30*b^7 + 14893056*a^31*b^6 - 11010048*a^32*b^5 + 4718592*a^33*b^4 - 1179648*a^34* b^3 + 131072*a^35*b^2 + (tan(e + f*x)*(262144*a^25*b^15 - 2883584*a^26*b^1 4 + 14155776*a^27*b^13 - 40370176*a^28*b^12 + 72089600*a^29*b^11 - 7785676 8*a^30*b^10 + 34603008*a^31*b^9 + 34603008*a^32*b^8 - 77856768*a^33*b^7 + 72089600*a^34*b^6 - 40370176*a^35*b^5 + 14155776*a^36*b^4 - 2883584*a^3...
\[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx=\int \frac {\cot \left (f x +e \right )^{6}}{\left (\tan \left (f x +e \right )^{2} b +a \right )^{3}}d x \] Input:
int(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x)
Output:
int(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x)