Integrand size = 25, antiderivative size = 163 \[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=-\frac {\left (8 a^2-4 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{3/2} f}+\frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(4 a-b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 a f}-\frac {\cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f} \]
-1/8*(8*a^2-4*a*b-b^2)*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))*(a-b)^(1/2)/f+1/8*(4*a-b)*c ot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2)/a/f-1/4*cot(f*x+e)^4*(a+b*tan(f*x+e)^ 2)^(1/2)/f
Time = 1.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\left (-8 a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 a \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )-\cot ^2(e+f x) \left (-4 a+b+2 a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}\right )}{8 a^{3/2} f} \]
((-8*a^2 + 4*a*b + b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt [a]*(8*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] - Cot [e + f*x]^2*(-4*a + b + 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2]))/( 8*a^(3/2)*f)
Time = 0.37 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4153, 354, 110, 27, 168, 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.
| \(\displaystyle \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \tan (e+f x)^2}}{\tan (e+f x)^5}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^5(e+f x) \sqrt {b \tan ^2(e+f x)+a}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x) \sqrt {b \tan ^2(e+f x)+a}}{\tan ^2(e+f x)+1}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\frac {1}{2} \int -\frac {\cot ^2(e+f x) \left (3 b \tan ^2(e+f x)+4 a-b\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{4} \int \frac {\cot ^2(e+f x) \left (3 b \tan ^2(e+f x)+4 a-b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\int \frac {\cot (e+f x) \left (8 a^2-4 b a-b^2+(4 a-b) b \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a}+\frac {(4 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\int \frac {\cot (e+f x) \left (8 a^2-4 b a-b^2+(4 a-b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}+\frac {(4 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\left (8 a^2-4 a b-b^2\right ) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-8 a (a-b) \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}+\frac {(4 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {2 \left (8 a^2-4 a b-b^2\right ) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {16 a (a-b) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{2 a}+\frac {(4 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {16 a \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )-\frac {2 \left (8 a^2-4 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {(4 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{2} \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
(-1/2*(Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]) + (((-2*(8*a^2 - 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + 16*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/(2*a) + ((4*a - b)*Co t[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/4)/(2*f)
3.3.98.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[(((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. \(2364\) vs. \(2(141)=282\).
Time = 1.05 (sec) , antiderivative size = 2365, normalized size of antiderivative = 14.51
-1/64/f/a^(7/2)/(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)/((-cos(f*x+e )+1)^2*csc(f*x+e)^2-1)^2)^(1/2)*((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)*(-8*(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*(-co s(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(-cos(f*x+e)+1)^6*a^(7/2)*(a-b)^(1/2)* csc(f*x+e)^6-2*b*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*c sc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*(-cos(f*x+e)+1)^6* a^(5/2)*(a-b)^(1/2)*csc(f*x+e)^6-32*a^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*(-c os(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))*(-cos(f*x+e)+1)^4*(a-b)^(1/2)*csc(f*x+ e)^4+16*(a-b)^(1/2)*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^3*b*(-cos(f*x+e)+1)^4*csc(f*x+e)^4+32*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^4*(-cos(f*x+e)+1)^4*(a-b)^(1/2)*csc(f*x+e)^4-16*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...
Time = 0.29 (sec) , antiderivative size = 729, normalized size of antiderivative = 4.47 \[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\left [\frac {8 \, \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 )^{4} - {\left (8 \, a^{2} - 4 \, 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 )^{4} + 2 \, {\left ({\left (4 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a^{2} f \tan \left (f x + e\right )^{4}}, \frac {16 \, 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 )^{4} - {\left (8 \, a^{2} - 4 \, 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 )^{4} + 2 \, {\left ({\left (4 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a^{2} f \tan \left (f x + e\right )^{4}}, \frac {4 \, \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 )^{4} + {\left (8 \, a^{2} - 4 \, 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 )^{4} + {\left ({\left (4 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a^{2} f \tan \left (f x + e\right )^{4}}, \frac {8 \, 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 )^{4} + {\left (8 \, a^{2} - 4 \, 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 )^{4} + {\left ({\left (4 \, a^{2} - a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a^{2} f \tan \left (f x + e\right )^{4}}\right ] \]
[1/16*(8*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)^4 - (8*a^2 - 4*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)^4 + 2*((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4), 1/16*( 16*a^2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b ))*tan(f*x + e)^4 - (8*a^2 - 4*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)^4 + 2*((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a ^2*f*tan(f*x + e)^4), 1/8*(4*sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 + 2*sqr t(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f *x + e)^4 + (8*a^2 - 4*a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + ((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqr t(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4), 1/8*(8*a^2*sqrt(-a + b)*a rctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + ( 8*a^2 - 4*a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a )*tan(f*x + e)^4 + ((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4)]
\[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \cot ^{5}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int { \sqrt {b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{5} \,d x } \]
Exception generated. \[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 11.20 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.33 \[ \int \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\mathrm {atanh}\left (\frac {3\,b^7\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{64\,\sqrt {a^3}\,\left (\frac {a\,b^5}{4}-\frac {11\,b^6}{32}+\frac {3\,b^7}{64\,a}+\frac {11\,b^8}{256\,a^2}+\frac {b^9}{256\,a^3}\right )}-\frac {11\,b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{32\,\sqrt {a^3}\,\left (\frac {b^5}{4}-\frac {11\,b^6}{32\,a}+\frac {3\,b^7}{64\,a^2}+\frac {11\,b^8}{256\,a^3}+\frac {b^9}{256\,a^4}\right )}+\frac {11\,b^8\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,\sqrt {a^3}\,\left (\frac {3\,b^7}{64}-\frac {11\,a\,b^6}{32}+\frac {a^2\,b^5}{4}+\frac {11\,b^8}{256\,a}+\frac {b^9}{256\,a^2}\right )}+\frac {b^9\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,\sqrt {a^3}\,\left (\frac {3\,a\,b^7}{64}+\frac {11\,b^8}{256}-\frac {11\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {b^9}{256\,a}\right )}+\frac {a\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{4\,\sqrt {a^3}\,\left (\frac {b^5}{4}-\frac {11\,b^6}{32\,a}+\frac {3\,b^7}{64\,a^2}+\frac {11\,b^8}{256\,a^3}+\frac {b^9}{256\,a^4}\right )}\right )\,\left (-8\,a^2+4\,a\,b+b^2\right )}{8\,f\,\sqrt {a^3}}-\frac {\mathrm {atanh}\left (\frac {b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a-b}}{4\,\left (\frac {7\,b^6}{32}-\frac {a\,b^5}{4}+\frac {b^7}{32\,a}\right )}+\frac {b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a-b}}{32\,\left (-\frac {a^2\,b^5}{4}+\frac {7\,a\,b^6}{32}+\frac {b^7}{32}\right )}\right )\,\sqrt {a-b}}{f}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (\frac {b^2}{8}+\frac {a\,b}{2}\right )-\frac {b\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}\,\left (4\,a-b\right )}{8\,a}}{f\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^2+a^2\,f-2\,a\,f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )} \]
(atanh((3*b^7*(a + b*tan(e + f*x)^2)^(1/2))/(64*(a^3)^(1/2)*((a*b^5)/4 - ( 11*b^6)/32 + (3*b^7)/(64*a) + (11*b^8)/(256*a^2) + b^9/(256*a^3))) - (11*b ^6*(a + b*tan(e + f*x)^2)^(1/2))/(32*(a^3)^(1/2)*(b^5/4 - (11*b^6)/(32*a) + (3*b^7)/(64*a^2) + (11*b^8)/(256*a^3) + b^9/(256*a^4))) + (11*b^8*(a + b *tan(e + f*x)^2)^(1/2))/(256*(a^3)^(1/2)*((3*b^7)/64 - (11*a*b^6)/32 + (a^ 2*b^5)/4 + (11*b^8)/(256*a) + b^9/(256*a^2))) + (b^9*(a + b*tan(e + f*x)^2 )^(1/2))/(256*(a^3)^(1/2)*((3*a*b^7)/64 + (11*b^8)/256 - (11*a^2*b^6)/32 + (a^3*b^5)/4 + b^9/(256*a))) + (a*b^5*(a + b*tan(e + f*x)^2)^(1/2))/(4*(a^ 3)^(1/2)*(b^5/4 - (11*b^6)/(32*a) + (3*b^7)/(64*a^2) + (11*b^8)/(256*a^3) + b^9/(256*a^4))))*(4*a*b - 8*a^2 + b^2))/(8*f*(a^3)^(1/2)) - (atanh((b^5* (a + b*tan(e + f*x)^2)^(1/2)*(a - b)^(1/2))/(4*((7*b^6)/32 - (a*b^5)/4 + b ^7/(32*a))) + (b^6*(a + b*tan(e + f*x)^2)^(1/2)*(a - b)^(1/2))/(32*((7*a*b ^6)/32 + b^7/32 - (a^2*b^5)/4)))*(a - b)^(1/2))/f - ((a + b*tan(e + f*x)^2 )^(1/2)*((a*b)/2 + b^2/8) - (b*(a + b*tan(e + f*x)^2)^(3/2)*(4*a - b))/(8* a))/(f*(a + b*tan(e + f*x)^2)^2 + a^2*f - 2*a*f*(a + b*tan(e + f*x)^2))