Integrand size = 25, antiderivative size = 215 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\left (8 a^2+12 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {b \left (4 a^2+3 a b-15 b^2\right )}{8 a^3 (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 a f \sqrt {a+b \tan ^2(e+f x)}} \]
-1/8*(8*a^2+12*a*b+15*b^2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(7/ 2)/f+arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/f+1/8*b*(4* a^2+3*a*b-15*b^2)/a^3/(a-b)/f/(a+b*tan(f*x+e)^2)^(1/2)+1/8*(4*a+5*b)*cot(f *x+e)^2/a^2/f/(a+b*tan(f*x+e)^2)^(1/2)-1/4*cot(f*x+e)^4/a/f/(a+b*tan(f*x+e )^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\frac {8 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (a \cot ^2(e+f x) \left (-4 a-5 b+2 a \cot ^2(e+f x)\right )-\left (8 a^2+12 a b+15 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \tan ^2(e+f x)}{a}\right )\right )}{8 a^3 (-a+b) f \sqrt {a+b \tan ^2(e+f x)}} \]
(8*a^3*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)] + ( a - b)*(a*Cot[e + f*x]^2*(-4*a - 5*b + 2*a*Cot[e + f*x]^2) - (8*a^2 + 12*a *b + 15*b^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a]))/( 8*a^3*(-a + b)*f*Sqrt[a + b*Tan[e + f*x]^2])
Time = 0.46 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4153, 354, 114, 27, 168, 27, 169, 27, 174, 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.
| \(\displaystyle \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^5 \left (a+b \tan (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^5(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (5 b \tan ^2(e+f x)+4 a+5 b\right )}{2 \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{2 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (5 b \tan ^2(e+f x)+4 a+5 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+12 b a+15 b^2+3 b (4 a+5 b) \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+12 b a+15 b^2+3 b (4 a+5 b) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2 b \left (4 a^2+3 a b-15 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 \int -\frac {\cot (e+f x) \left (b \left (4 a^2+3 b a-15 b^2\right ) \tan ^2(e+f x)+(a-b) \left (8 a^2+12 b a+15 b^2\right )\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {\cot (e+f x) \left (b \left (4 a^2+3 b a-15 b^2\right ) \tan ^2(e+f x)+(a-b) \left (8 a^2+12 b a+15 b^2\right )\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}+\frac {2 b \left (4 a^2+3 a b-15 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {(a-b) \left (8 a^2+12 a b+15 b^2\right ) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-8 a^3 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}+\frac {2 b \left (4 a^2+3 a b-15 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 (a-b) \left (8 a^2+12 a b+15 b^2\right ) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {16 a^3 \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{a (a-b)}+\frac {2 b \left (4 a^2+3 a b-15 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2 b \left (4 a^2+3 a b-15 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}+\frac {\frac {16 a^3 \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b) \left (8 a^2+12 a b+15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}}{2 a}-\frac {(4 a+5 b) \cot (e+f x)}{a \sqrt {a+b \tan ^2(e+f x)}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
(-1/2*Cot[e + f*x]^2/(a*Sqrt[a + b*Tan[e + f*x]^2]) - (-(((4*a + 5*b)*Cot[ e + f*x])/(a*Sqrt[a + b*Tan[e + f*x]^2])) - (((-2*(a - b)*(8*a^2 + 12*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (16*a^3*A rcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/(a*(a - b)) + (2*b*(4*a^2 + 3*a*b - 15*b^2))/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2]))/(2 *a))/(4*a))/(2*f)
3.4.38.3.1 Defintions of rubi rules used
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_))^(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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(25556\) vs. \(2(189)=378\).
Time = 1.78 (sec) , antiderivative size = 25557, normalized size of antiderivative = 118.87
Time = 0.35 (sec) , antiderivative size = 1522, normalized size of antiderivative = 7.08 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/16*(8*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(a - b)*log((b*t an(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f *x + e)^2 + 1)) - ((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan (f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt( a) + 2*a)/tan(f*x + e)^2) + 2*(2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^ 3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b ^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^ 5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/16*(16*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(-a + b)*a rctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + ((8*a^4*b - 4*a^ 3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a ^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e )^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(2*a ^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan (f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sq rt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/8*(((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*...
\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
Time = 13.70 (sec) , antiderivative size = 2118, normalized size of antiderivative = 9.85 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
(atan((((((a + b*tan(e + f*x)^2)^(1/2)*(230400*a^9*b^11*f^3 - 783360*a^10* b^10*f^3 + 854016*a^11*b^9*f^3 - 387072*a^12*b^8*f^3 + 480256*a^13*b^7*f^3 - 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 32768 0*a^17*b^3*f^3 - 131072*a^18*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(638976*a^13 *b^9*f^4 - 122880*a^12*b^10*f^4 - 1318912*a^14*b^8*f^4 + 1376256*a^15*b^7* f^4 - 794624*a^16*b^6*f^4 + 311296*a^17*b^5*f^4 - 122880*a^18*b^4*f^4 + 32 768*a^19*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144 *a^15*b^8*f^5 - 1835008*a^16*b^7*f^5 + 5242880*a^17*b^6*f^5 - 7864320*a^18 *b^5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21*b^2*f ^5))/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3 ) + ((((a + b*tan(e + f*x)^2)^(1/2)*(230400*a^9*b^11*f^3 - 783360*a^10*b^1 0*f^3 + 854016*a^11*b^9*f^3 - 387072*a^12*b^8*f^3 + 480256*a^13*b^7*f^3 - 680960*a^14*b^6*f^3 + 352256*a^15*b^5*f^3 - 262144*a^16*b^4*f^3 + 327680*a ^17*b^3*f^3 - 131072*a^18*b^2*f^3))/2 + (((a - b)^3)^(1/2)*(122880*a^12*b^ 10*f^4 - 638976*a^13*b^9*f^4 + 1318912*a^14*b^8*f^4 - 1376256*a^15*b^7*f^4 + 794624*a^16*b^6*f^4 - 311296*a^17*b^5*f^4 + 122880*a^18*b^4*f^4 - 32768 *a^19*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b)^3)^(1/2)*(262144*a^ 15*b^8*f^5 - 1835008*a^16*b^7*f^5 + 5242880*a^17*b^6*f^5 - 7864320*a^18*b^ 5*f^5 + 6553600*a^19*b^4*f^5 - 2883584*a^20*b^3*f^5 + 524288*a^21*b^2*f^5) )/(4*f*(a - b)^3)))/(2*f*(a - b)^3))*((a - b)^3)^(1/2)*1i)/(f*(a - b)^3...