Integrand size = 23, antiderivative size = 285 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {x}{a^3}+\frac {b^{7/2} \left (99 a^2+44 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{11/2} f}-\frac {\left (8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right ) \cot (e+f x)}{8 a^2 (a+b)^5 f}+\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^4 f}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{40 a^2 (a+b)^3 f}-\frac {b \cot ^5(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )} \] Output:
-x/a^3+1/8*b^(7/2)*(99*a^2+44*a*b+8*b^2)*arctan(b^(1/2)*tan(f*x+e)/(a+b)^( 1/2))/a^3/(a+b)^(11/2)/f-1/8*(8*a^4+40*a^3*b+80*a^2*b^2-19*a*b^3-4*b^4)*co t(f*x+e)/a^2/(a+b)^5/f+1/24*(8*a^3+32*a^2*b-51*a*b^2-12*b^3)*cot(f*x+e)^3/ a^2/(a+b)^4/f-1/40*(8*a^2-75*a*b-20*b^2)*cot(f*x+e)^5/a^2/(a+b)^3/f-1/4*b* cot(f*x+e)^5/a/(a+b)/f/(a+b+b*tan(f*x+e)^2)^2-1/8*b*(13*a+4*b)*cot(f*x+e)^ 5/a^2/(a+b)^2/f/(a+b+b*tan(f*x+e)^2)
Result contains complex when optimal does not.
Time = 9.91 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.92 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^6(e+f x) \left (-\frac {120 x (a+2 b+a \cos (2 (e+f x)))^2}{a^3}+\frac {8 (11 a+26 b) (a+2 b+a \cos (2 (e+f x)))^2 \cot (e) \csc ^2(e+f x)}{(a+b)^4 f}-\frac {24 (a+2 b+a \cos (2 (e+f x)))^2 \cot (e) \csc ^4(e+f x)}{(a+b)^3 f}-\frac {15 b^4 \left (99 a^2+44 a b+8 b^2\right ) \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (a+2 b+a \cos (2 (e+f x)))^2 (\cos (2 e)-i \sin (2 e))}{a^3 (a+b)^{11/2} f \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {8 \left (23 a^2+106 a b+173 b^2\right ) (a+2 b+a \cos (2 (e+f x)))^2 \csc (e) \csc (e+f x) \sin (f x)}{(a+b)^5 f}-\frac {8 (11 a+26 b) (a+2 b+a \cos (2 (e+f x)))^2 \csc (e) \csc ^3(e+f x) \sin (f x)}{(a+b)^4 f}+\frac {24 (a+2 b+a \cos (2 (e+f x)))^2 \csc (e) \csc ^5(e+f x) \sin (f x)}{(a+b)^3 f}+\frac {60 b^5 \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{a^3 (a+b)^4 f}-\frac {15 b^4 (a+2 b+a \cos (2 (e+f x))) \sec (2 e) \left (\left (21 a^2+52 a b+16 b^2\right ) \sin (2 e)-3 a (7 a+2 b) \sin (2 f x)\right )}{a^3 (a+b)^5 f}\right )}{960 \left (a+b \sec ^2(e+f x)\right )^3} \] Input:
Integrate[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]
Output:
((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*((-120*x*(a + 2*b + a*Cos[2 *(e + f*x)])^2)/a^3 + (8*(11*a + 26*b)*(a + 2*b + a*Cos[2*(e + f*x)])^2*Co t[e]*Csc[e + f*x]^2)/((a + b)^4*f) - (24*(a + 2*b + a*Cos[2*(e + f*x)])^2* Cot[e]*Csc[e + f*x]^4)/((a + b)^3*f) - (15*b^4*(99*a^2 + 44*a*b + 8*b^2)*A rcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[ 2*(e + f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(a^3*(a + b)^(11/2)*f*Sqrt[b*(Cos [e] - I*Sin[e])^4]) + (8*(23*a^2 + 106*a*b + 173*b^2)*(a + 2*b + a*Cos[2*( e + f*x)])^2*Csc[e]*Csc[e + f*x]*Sin[f*x])/((a + b)^5*f) - (8*(11*a + 26*b )*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e]*Csc[e + f*x]^3*Sin[f*x])/((a + b )^4*f) + (24*(a + 2*b + a*Cos[2*(e + f*x)])^2*Csc[e]*Csc[e + f*x]^5*Sin[f* x])/((a + b)^3*f) + (60*b^5*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/ (a^3*(a + b)^4*f) - (15*b^4*(a + 2*b + a*Cos[2*(e + f*x)])*Sec[2*e]*((21*a ^2 + 52*a*b + 16*b^2)*Sin[2*e] - 3*a*(7*a + 2*b)*Sin[2*f*x]))/(a^3*(a + b) ^5*f)))/(960*(a + b*Sec[e + f*x]^2)^3)
Time = 0.68 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4629, 2075, 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.
| \(\displaystyle \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^6 \left (a+b \sec (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (a+b \left (\tan ^2(e+f x)+1\right )\right )^3}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )^3}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^6(e+f x) \left (-9 b \tan ^2(e+f x)+4 a-5 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cot ^6(e+f x) \left (8 a^2-75 b a-20 b^2-7 b (13 a+4 b) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {5 \cot ^4(e+f x) \left (8 a^3+32 b a^2-51 b^2 a-12 b^3+b \left (8 a^2-75 b a-20 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{5 (a+b)}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\cot ^4(e+f x) \left (8 a^3+32 b a^2-51 b^2 a-12 b^3+b \left (8 a^2-75 b a-20 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {3 \cot ^2(e+f x) \left (8 a^4+40 b a^3+80 b^2 a^2-19 b^3 a-4 b^4+b \left (8 a^3+32 b a^2-51 b^2 a-12 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{3 (a+b)}-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {\cot ^2(e+f x) \left (8 a^4+40 b a^3+80 b^2 a^2-19 b^3 a-4 b^4+b \left (8 a^3+32 b a^2-51 b^2 a-12 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\int \frac {8 a^5+48 b a^4+120 b^2 a^3+160 b^3 a^2+21 b^4 a+4 b^5+b \left (8 a^4+40 b a^3+80 b^2 a^2-19 b^3 a-4 b^4\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a+b}-\frac {\left (8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\frac {8 (a+b)^5 \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a}-\frac {b^4 \left (99 a^2+44 a b+8 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a+b}-\frac {\left (8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {-\frac {-\frac {-\frac {\frac {8 (a+b)^5 \arctan (\tan (e+f x))}{a}-\frac {b^4 \left (99 a^2+44 a b+8 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a+b}-\frac {\left (8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}}{a+b}-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\left (8 a^2-75 a b-20 b^2\right ) \cot ^5(e+f x)}{5 (a+b)}-\frac {-\frac {\left (8 a^3+32 a^2 b-51 a b^2-12 b^3\right ) \cot ^3(e+f x)}{3 (a+b)}-\frac {-\frac {\frac {8 (a+b)^5 \arctan (\tan (e+f x))}{a}-\frac {b^{7/2} \left (99 a^2+44 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^4+40 a^3 b+80 a^2 b^2-19 a b^3-4 b^4\right ) \cot (e+f x)}{a+b}}{a+b}}{a+b}}{2 a (a+b)}-\frac {b (13 a+4 b) \cot ^5(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^5(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\) |
Input:
Int[Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]
Output:
(-1/4*(b*Cot[e + f*x]^5)/(a*(a + b)*(a + b + b*Tan[e + f*x]^2)^2) + ((-1/5 *((8*a^2 - 75*a*b - 20*b^2)*Cot[e + f*x]^5)/(a + b) - (-1/3*((8*a^3 + 32*a ^2*b - 51*a*b^2 - 12*b^3)*Cot[e + f*x]^3)/(a + b) - (-(((8*(a + b)^5*ArcTa n[Tan[e + f*x]])/a - (b^(7/2)*(99*a^2 + 44*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Ta n[e + f*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b)) - ((8*a^4 + 40*a^3*b + 80*a^2*b^2 - 19*a*b^3 - 4*b^4)*Cot[e + f*x])/(a + b))/(a + b))/(a + b))/( 2*a*(a + b)) - (b*(13*a + 4*b)*Cot[e + f*x]^5)/(2*a*(a + b)*(a + b + 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[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f _.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/f Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 )), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
Time = 45.07 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{5 \left (a +b \right )^{3} \tan \left (f x +e \right )^{5}}-\frac {-a -4 b}{3 \left (a +b \right )^{4} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+5 a b +10 b^{2}}{\left (a +b \right )^{5} \tan \left (f x +e \right )}+\frac {b^{4} \left (\frac {\left (\frac {19}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (21 a^{2}+25 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (99 a^{2}+44 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{5}}}{f}\) | \(199\) |
| default | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{5 \left (a +b \right )^{3} \tan \left (f x +e \right )^{5}}-\frac {-a -4 b}{3 \left (a +b \right )^{4} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+5 a b +10 b^{2}}{\left (a +b \right )^{5} \tan \left (f x +e \right )}+\frac {b^{4} \left (\frac {\left (\frac {19}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (21 a^{2}+25 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (99 a^{2}+44 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{5}}}{f}\) | \(199\) |
| risch | \(\text {Expression too large to display}\) | \(1412\) |
Input:
int(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/f*(-1/a^3*arctan(tan(f*x+e))-1/5/(a+b)^3/tan(f*x+e)^5-1/3*(-a-4*b)/(a+b) ^4/tan(f*x+e)^3-(a^2+5*a*b+10*b^2)/(a+b)^5/tan(f*x+e)+b^4/a^3/(a+b)^5*(((1 9/8*a^2*b+1/2*a*b^2)*tan(f*x+e)^3+1/8*a*(21*a^2+25*a*b+4*b^2)*tan(f*x+e))/ (a+b+b*tan(f*x+e)^2)^2+1/8*(99*a^2+44*a*b+8*b^2)/((a+b)*b)^(1/2)*arctan(b* tan(f*x+e)/((a+b)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (265) = 530\).
Time = 0.23 (sec) , antiderivative size = 2229, normalized size of antiderivative = 7.82 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")
Output:
[-1/480*(4*(184*a^7 + 848*a^6*b + 1384*a^5*b^2 + 315*a^3*b^4 + 90*a^2*b^5) *cos(f*x + e)^9 - 4*(280*a^7 + 1032*a^6*b + 864*a^5*b^2 - 2768*a^4*b^3 + 9 45*a^3*b^4 - 15*a^2*b^5 - 60*a*b^6)*cos(f*x + e)^7 + 4*(120*a^7 + 40*a^6*b - 1416*a^5*b^2 - 4272*a^4*b^3 + 2329*a^3*b^4 - 585*a^2*b^5 - 180*a*b^6)*c os(f*x + e)^5 + 20*(48*a^6*b + 184*a^5*b^2 + 200*a^4*b^3 - 575*a^3*b^4 + 1 53*a^2*b^5 + 36*a*b^6)*cos(f*x + e)^3 - 15*((99*a^4*b^3 + 44*a^3*b^4 + 8*a ^2*b^5)*cos(f*x + e)^8 + 99*a^2*b^5 + 44*a*b^6 + 8*b^7 - 2*(99*a^4*b^3 - 5 5*a^3*b^4 - 36*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^6 + (99*a^4*b^3 - 352*a^3*b ^4 - 69*a^2*b^5 + 12*a*b^6 + 8*b^7)*cos(f*x + e)^4 + 2*(99*a^3*b^4 - 55*a^ 2*b^5 - 36*a*b^6 - 8*b^7)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a *b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b)) *sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*si n(f*x + e) + 60*(8*a^5*b^2 + 40*a^4*b^3 + 80*a^3*b^4 - 19*a^2*b^5 - 4*a*b^ 6)*cos(f*x + e) + 480*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^ 4 + a^2*b^5)*f*x*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*x*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a ^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*x*cos(f*x + e)^4 + 2*(a^6 *b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*x*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f*x)...
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(cot(f*x+e)**6/(a+b*sec(f*x+e)**2)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.82 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {15 \, {\left (99 \, a^{2} b^{4} + 44 \, a b^{5} + 8 \, b^{6}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {15 \, {\left (8 \, a^{4} b^{2} + 40 \, a^{3} b^{3} + 80 \, a^{2} b^{4} - 19 \, a b^{5} - 4 \, b^{6}\right )} \tan \left (f x + e\right )^{8} + 5 \, {\left (48 \, a^{5} b + 280 \, a^{4} b^{2} + 680 \, a^{3} b^{3} + 385 \, a^{2} b^{4} - 75 \, a b^{5} - 12 \, b^{6}\right )} \tan \left (f x + e\right )^{6} + 24 \, a^{6} + 96 \, a^{5} b + 144 \, a^{4} b^{2} + 96 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 8 \, {\left (15 \, a^{6} + 95 \, a^{5} b + 258 \, a^{4} b^{2} + 291 \, a^{3} b^{3} + 113 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{4} - 8 \, {\left (5 \, a^{6} + 29 \, a^{5} b + 57 \, a^{4} b^{2} + 47 \, a^{3} b^{3} + 14 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{7} b^{2} + 5 \, a^{6} b^{3} + 10 \, a^{5} b^{4} + 10 \, a^{4} b^{5} + 5 \, a^{3} b^{6} + a^{2} b^{7}\right )} \tan \left (f x + e\right )^{9} + 2 \, {\left (a^{8} b + 6 \, a^{7} b^{2} + 15 \, a^{6} b^{3} + 20 \, a^{5} b^{4} + 15 \, a^{4} b^{5} + 6 \, a^{3} b^{6} + a^{2} b^{7}\right )} \tan \left (f x + e\right )^{7} + {\left (a^{9} + 7 \, a^{8} b + 21 \, a^{7} b^{2} + 35 \, a^{6} b^{3} + 35 \, a^{5} b^{4} + 21 \, a^{4} b^{5} + 7 \, a^{3} b^{6} + a^{2} b^{7}\right )} \tan \left (f x + e\right )^{5}} - \frac {120 \, {\left (f x + e\right )}}{a^{3}}}{120 \, f} \] Input:
integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")
Output:
1/120*(15*(99*a^2*b^4 + 44*a*b^5 + 8*b^6)*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*sq rt((a + b)*b)) - (15*(8*a^4*b^2 + 40*a^3*b^3 + 80*a^2*b^4 - 19*a*b^5 - 4*b ^6)*tan(f*x + e)^8 + 5*(48*a^5*b + 280*a^4*b^2 + 680*a^3*b^3 + 385*a^2*b^4 - 75*a*b^5 - 12*b^6)*tan(f*x + e)^6 + 24*a^6 + 96*a^5*b + 144*a^4*b^2 + 9 6*a^3*b^3 + 24*a^2*b^4 + 8*(15*a^6 + 95*a^5*b + 258*a^4*b^2 + 291*a^3*b^3 + 113*a^2*b^4)*tan(f*x + e)^4 - 8*(5*a^6 + 29*a^5*b + 57*a^4*b^2 + 47*a^3* b^3 + 14*a^2*b^4)*tan(f*x + e)^2)/((a^7*b^2 + 5*a^6*b^3 + 10*a^5*b^4 + 10* a^4*b^5 + 5*a^3*b^6 + a^2*b^7)*tan(f*x + e)^9 + 2*(a^8*b + 6*a^7*b^2 + 15* a^6*b^3 + 20*a^5*b^4 + 15*a^4*b^5 + 6*a^3*b^6 + a^2*b^7)*tan(f*x + e)^7 + (a^9 + 7*a^8*b + 21*a^7*b^2 + 35*a^6*b^3 + 35*a^5*b^4 + 21*a^4*b^5 + 7*a^3 *b^6 + a^2*b^7)*tan(f*x + e)^5) - 120*(f*x + e)/a^3)/f
Time = 0.41 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.36 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {15 \, {\left (99 \, a^{2} b^{4} + 44 \, a b^{5} + 8 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5}\right )} \sqrt {a b + b^{2}}} + \frac {15 \, {\left (19 \, a b^{5} \tan \left (f x + e\right )^{3} + 4 \, b^{6} \tan \left (f x + e\right )^{3} + 21 \, a^{2} b^{4} \tan \left (f x + e\right ) + 25 \, a b^{5} \tan \left (f x + e\right ) + 4 \, b^{6} \tan \left (f x + e\right )\right )}}{{\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac {120 \, {\left (f x + e\right )}}{a^{3}} - \frac {8 \, {\left (15 \, a^{2} \tan \left (f x + e\right )^{4} + 75 \, a b \tan \left (f x + e\right )^{4} + 150 \, b^{2} \tan \left (f x + e\right )^{4} - 5 \, a^{2} \tan \left (f x + e\right )^{2} - 25 \, a b \tan \left (f x + e\right )^{2} - 20 \, b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}\right )}}{{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \tan \left (f x + e\right )^{5}}}{120 \, f} \] Input:
integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")
Output:
1/120*(15*(99*a^2*b^4 + 44*a*b^5 + 8*b^6)*(pi*floor((f*x + e)/pi + 1/2)*sg n(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))/((a^8 + 5*a^7*b + 10*a^6*b^ 2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*sqrt(a*b + b^2)) + 15*(19*a*b^5*tan( f*x + e)^3 + 4*b^6*tan(f*x + e)^3 + 21*a^2*b^4*tan(f*x + e) + 25*a*b^5*tan (f*x + e) + 4*b^6*tan(f*x + e))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*(b*tan(f*x + e)^2 + a + b)^2) - 120*(f*x + e)/a^3 - 8*(15*a^2*tan(f*x + e)^4 + 75*a*b*tan(f*x + e)^4 + 150*b^2*tan(f*x + e)^4 - 5*a^2*tan(f*x + e)^2 - 25*a*b*tan(f*x + e)^2 - 20*b^2*tan(f*x + e)^2 + 3*a^2 + 6*a*b + 3*b^2)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*tan(f*x + e)^5))/f
Time = 23.61 (sec) , antiderivative size = 7460, normalized size of antiderivative = 26.18 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(cot(e + f*x)^6/(a + b/cos(e + f*x)^2)^3,x)
Output:
atan(((((65536*a^10*b^27 + 1654784*a^11*b^26 + 21954560*a^12*b^25 + 194478 080*a^13*b^24 + 1247936512*a^14*b^23 + 6060916736*a^15*b^22 + 22968795136* a^16*b^21 + 69506170880*a^17*b^20 + 170976215040*a^18*b^19 + 346596343808* a^19*b^18 + 585044721664*a^20*b^17 + 828584034304*a^21*b^16 + 989821665280 *a^22*b^15 + 1000564490240*a^23*b^14 + 856970493952*a^24*b^13 + 6215385743 36*a^25*b^12 + 380751118336*a^26*b^11 + 196065116160*a^27*b^10 + 842304716 80*a^28*b^9 + 29853974528*a^29*b^8 + 8588754944*a^30*b^7 + 1957904384*a^31 *b^6 + 340787200*a^32*b^5 + 42598400*a^33*b^4 + 3407872*a^34*b^3 + 131072* a^35*b^2 + (tan(e + f*x)*(524288*a^12*b^28 + 13369344*a^13*b^27 + 16384000 0*a^14*b^26 + 1284505600*a^15*b^25 + 7235174400*a^16*b^24 + 31171543040*a^ 17*b^23 + 106779115520*a^18*b^22 + 298450944000*a^19*b^21 + 693069414400*a ^20*b^20 + 1354635673600*a^21*b^19 + 2249325281280*a^22*b^18 + 31938471526 40*a^23*b^17 + 3894935552000*a^24*b^16 + 4089682329600*a^25*b^15 + 3700188 774400*a^26*b^14 + 2882252308480*a^27*b^13 + 1927993098240*a^28*b^12 + 110 2610432000*a^29*b^11 + 535553638400*a^30*b^10 + 218864025600*a^31*b^9 + 74 281123840*a^32*b^8 + 20559953920*a^33*b^7 + 4521984000*a^34*b^6 + 76021760 0*a^35*b^5 + 91750400*a^36*b^4 + 7077888*a^37*b^3 + 262144*a^38*b^2)*1i)/( 2*a^3))*1i)/(2*a^3) + tan(e + f*x)*(131072*a^6*b^28 + 3342336*a^7*b^27 + 4 0960000*a^8*b^26 + 319234048*a^9*b^25 + 1768817664*a^10*b^24 + 7390051328* a^11*b^23 + 24132297728*a^12*b^22 + 63100984320*a^13*b^21 + 13447268454...
Time = 14.79 (sec) , antiderivative size = 2455, normalized size of antiderivative = 8.61 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x)
Output:
(1485*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sq rt(b))*sin(e + f*x)**9*a**4*b**3 + 660*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**9*a**3*b**4 + 120*sq rt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*s in(e + f*x)**9*a**2*b**5 - 2970*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan( (e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**7*a**4*b**3 - 4290*sqrt(b)* sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**7*a**3*b**4 - 1560*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f *x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**7*a**2*b**5 - 240*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)** 7*a*b**6 + 1485*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - s qrt(a))/sqrt(b))*sin(e + f*x)**5*a**4*b**3 + 3630*sqrt(b)*sqrt(a + b)*atan ((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**5*a**3*b* *4 + 2925*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a) )/sqrt(b))*sin(e + f*x)**5*a**2*b**5 + 900*sqrt(b)*sqrt(a + b)*atan((sqrt( a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**5*a*b**6 + 120*s qrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))* sin(e + f*x)**5*b**7 + 1485*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f*x)**9*a**4*b**3 + 660*sqrt(b)*sqrt( a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f...