Integrand size = 25, antiderivative size = 116 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a f} \]
1/2*(2*a+b)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f-arctanh((a +b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f/(a-b)^(1/2)-1/2*cot(f*x+e)^2*(a+b*ta n(f*x+e)^2)^(1/2)/a/f
Time = 0.83 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\left (2 a^2-a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (-2 a \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+(-a+b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )}{2 a^{3/2} (a-b) f} \]
((2*a^2 - a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a] *(-2*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (-a + b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]))/(2*a^(3/2)*(a - b)*f)
Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4153, 354, 114, 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.
| \(\displaystyle \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^3 \sqrt {a+b \tan (e+f x)^2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot (e+f x) \left (b \tan ^2(e+f x)+2 a+b\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot (e+f x) \left (b \tan ^2(e+f x)+2 a+b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {-\frac {(2 a+b) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-2 a \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {2 (2 a+b) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {4 a \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\) |
(-1/2*((-2*(2*a + b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (4*a*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/a - ( Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/(2*f)
3.4.24.3.1 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[(((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. \(1179\) vs. \(2(98)=196\).
Time = 1.14 (sec) , antiderivative size = 1180, normalized size of antiderivative = 10.17
-1/8/f/a^(5/2)/(a-b)^(1/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x +e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)/((a*(-co s(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f* x+e)+1)^2*csc(f*x+e)^2+a)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^2)^(1/2)/((-c os(f*x+e)+1)^2*csc(f*x+e)^2-1)/(-cos(f*x+e)+1)^2*((a*(-cos(f*x+e)+1)^4*csc (f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x +e)^2+a)^(1/2)*a^(3/2)*(-cos(f*x+e)+1)^2*(a-b)^(1/2)-8*ln(4*(-a*(-cos(f*x+ e)+1)^2*csc(f*x+e)^2+b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a-b)^(1/2)*(a*(-cos (f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x +e)+1)^2*csc(f*x+e)^2+a)^(1/2)+a-b)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2+1))*a^ (5/2)*(-cos(f*x+e)+1)^2-4*ln((a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a*(-cos(f* x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e) +1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*a^2*(-cos(f*x+e)+1)^2* (a-b)^(1/2)+4*ln(2/(-cos(f*x+e)+1)^2*(-a*(-cos(f*x+e)+1)^2+2*b*(-cos(f*x+e )+1)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^ 2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f *x+e)^2))*a^2*(-cos(f*x+e)+1)^2*(a-b)^(1/2)-2*ln((a*(-cos(f*x+e)+1)^2*csc( f*x+e)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e )^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*a* (-cos(f*x+e)+1)^2*(a-b)^(1/2)*b+2*ln(2/(-cos(f*x+e)+1)^2*(-a*(-cos(f*x+...
Time = 0.32 (sec) , antiderivative size = 697, normalized size of antiderivative = 6.01 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\left [\frac {2 \, \sqrt {a - b} a^{2} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} + {\left (2 \, a^{2} - a b - b^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{4 \, {\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{2}}, -\frac {4 \, a^{2} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{2} - {\left (2 \, a^{2} - a b - b^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) \tan \left (f x + e\right )^{2} + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{4 \, {\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{2}}, \frac {\sqrt {a - b} a^{2} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} - {\left (2 \, a^{2} - a b - b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{2} - \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{2 \, {\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{2}}, -\frac {2 \, a^{2} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{2} + {\left (2 \, a^{2} - a b - b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{2} + \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{2 \, {\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{2}}\right ] \]
[1/4*(2*sqrt(a - b)*a^2*log((b*tan(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))*tan(f*x + e)^2 + (2*a^2 - a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sq rt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a) *(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), -1/4*(4*a^2*sqrt(-a + b)*a rctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 - ( 2*a^2 - a*b - b^2)*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)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e )^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), 1/2*(sqrt(a - b)*a ^2*log((b*tan(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))*tan(f*x + e)^2 - (2*a^2 - a*b - b^2)*sqrt(-a)*a rctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 - sqrt(b*tan(f *x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), -1/2*(2*a^2* sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan( f*x + e)^2 + (2*a^2 - a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a )*sqrt(-a)/a)*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a ^3 - a^2*b)*f*tan(f*x + e)^2)]
\[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {\cot ^{3}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{3}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{3}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
Time = 0.44 (sec) , antiderivative size = 830, normalized size of antiderivative = 7.16 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\mathrm {atanh}\left (\frac {b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{4\,\sqrt {a^3}\,\left (\frac {3\,a\,b^4}{2}+\frac {5\,b^5}{4}+\frac {b^6}{4\,a}\right )}+\frac {3\,b^4\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{2\,\sqrt {a^3}\,\left (\frac {3\,b^4}{2\,a}+\frac {5\,b^5}{4\,a^2}+\frac {b^6}{4\,a^3}\right )}+\frac {5\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{4\,\sqrt {a^3}\,\left (\frac {3\,b^4}{2}+\frac {5\,b^5}{4\,a}+\frac {b^6}{4\,a^2}\right )}\right )\,\left (2\,a+b\right )}{2\,f\,\sqrt {a^3}}-\frac {b\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{2\,a\,\left (f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )-a\,f\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\frac {2\,a^2\,b^3\,f^2+2\,a\,b^4\,f^2}{2\,a^2\,f^3}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (16\,a^2\,b^3\,f^2-32\,a^3\,b^2\,f^2\right )}{8\,a^2\,f^3\,\sqrt {a-b}}}{2\,f\,\sqrt {a-b}}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (8\,a^2\,b^2+4\,a\,b^3+b^4\right )}{4\,a^2\,f^2}\right )\,1{}\mathrm {i}}{f\,\sqrt {a-b}}-\frac {\left (\frac {\frac {2\,a^2\,b^3\,f^2+2\,a\,b^4\,f^2}{2\,a^2\,f^3}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (16\,a^2\,b^3\,f^2-32\,a^3\,b^2\,f^2\right )}{8\,a^2\,f^3\,\sqrt {a-b}}}{2\,f\,\sqrt {a-b}}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (8\,a^2\,b^2+4\,a\,b^3+b^4\right )}{4\,a^2\,f^2}\right )\,1{}\mathrm {i}}{f\,\sqrt {a-b}}}{\frac {\frac {\frac {2\,a^2\,b^3\,f^2+2\,a\,b^4\,f^2}{2\,a^2\,f^3}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (16\,a^2\,b^3\,f^2-32\,a^3\,b^2\,f^2\right )}{8\,a^2\,f^3\,\sqrt {a-b}}}{2\,f\,\sqrt {a-b}}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (8\,a^2\,b^2+4\,a\,b^3+b^4\right )}{4\,a^2\,f^2}}{f\,\sqrt {a-b}}+\frac {\frac {\frac {2\,a^2\,b^3\,f^2+2\,a\,b^4\,f^2}{2\,a^2\,f^3}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (16\,a^2\,b^3\,f^2-32\,a^3\,b^2\,f^2\right )}{8\,a^2\,f^3\,\sqrt {a-b}}}{2\,f\,\sqrt {a-b}}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (8\,a^2\,b^2+4\,a\,b^3+b^4\right )}{4\,a^2\,f^2}}{f\,\sqrt {a-b}}-\frac {\frac {b^4}{2}+a\,b^3}{a^2\,f^3}}\right )\,1{}\mathrm {i}}{f\,\sqrt {a-b}} \]
(atan((((((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^3) - ((a + b*tan(e + f*x) ^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2 *f*(a - b)^(1/2)) - ((a + b*tan(e + f*x)^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b ^2))/(4*a^2*f^2))*1i)/(f*(a - b)^(1/2)) - ((((2*a*b^4*f^2 + 2*a^2*b^3*f^2) /(2*a^2*f^3) + ((a + b*tan(e + f*x)^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2* f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*tan(e + f*x )^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b^2))/(4*a^2*f^2))*1i)/(f*(a - b)^(1/2)) )/((((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^3) - ((a + b*tan(e + f*x)^2)^( 1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) - ((a + b*tan(e + f*x)^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b^2))/ (4*a^2*f^2))/(f*(a - b)^(1/2)) + (((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^ 3) + ((a + b*tan(e + f*x)^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a ^2*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*tan(e + f*x)^2)^(1/2) *(4*a*b^3 + b^4 + 8*a^2*b^2))/(4*a^2*f^2))/(f*(a - b)^(1/2)) - (a*b^3 + b^ 4/2)/(a^2*f^3)))*1i)/(f*(a - b)^(1/2)) - (b*(a + b*tan(e + f*x)^2)^(1/2))/ (2*a*(f*(a + b*tan(e + f*x)^2) - a*f)) + (atanh((b^6*(a + b*tan(e + f*x)^2 )^(1/2))/(4*(a^3)^(1/2)*((3*a*b^4)/2 + (5*b^5)/4 + b^6/(4*a))) + (3*b^4*(a + b*tan(e + f*x)^2)^(1/2))/(2*(a^3)^(1/2)*((3*b^4)/(2*a) + (5*b^5)/(4*a^2 ) + b^6/(4*a^3))) + (5*b^5*(a + b*tan(e + f*x)^2)^(1/2))/(4*(a^3)^(1/2)*(( 3*b^4)/2 + (5*b^5)/(4*a) + b^6/(4*a^2))))*(2*a + b))/(2*f*(a^3)^(1/2))