Integrand size = 40, antiderivative size = 179 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=-\frac {2 g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c f}+\frac {g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {a} c f}+\frac {g^2 \cot (e+f x) \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}{a c f} \]
-2*g^(5/2)*arctanh(a^(1/2)*g^(1/2)*tan(f*x+e)/(g*sec(f*x+e))^(1/2)/(a+a*se c(f*x+e))^(1/2))/c/f/a^(1/2)+1/2*g^(5/2)*arctanh(1/2*a^(1/2)*g^(1/2)*tan(f *x+e)*2^(1/2)/(g*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2))/c/f*2^(1/2)/a^( 1/2)+g^2*cot(f*x+e)*(g*sec(f*x+e))^(1/2)*(a+a*sec(f*x+e))^(1/2)/a/c/f
Time = 3.07 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.83 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=-\frac {(g \sec (e+f x))^{5/2} \sqrt {1+\sec (e+f x)} \sin ^3(e+f x) \left (8 \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)}+\left (16 \log (1+\sec (e+f x))-16 \log \left (\sqrt {\sec (e+f x)}+\sec ^{\frac {3}{2}}(e+f x)+\sqrt {1+\sec (e+f x)} \sqrt {\tan ^2(e+f x)}\right )+\sqrt {2} \left (\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)-2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {\tan ^2(e+f x)}\right )-\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)+2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {\tan ^2(e+f x)}\right )\right )\right ) \sqrt {\tan ^2(e+f x)}\right )}{8 c f (-1+\cos (e+f x)) (1+\cos (e+f x))^2 (-1+\sec (e+f x)) \sec ^{\frac {5}{2}}(e+f x) \sqrt {a (1+\sec (e+f x))}} \]
-1/8*((g*Sec[e + f*x])^(5/2)*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]^3*(8*Sqrt [Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]] + (16*Log[1 + Sec[e + f*x]] - 16*Log [Sqrt[Sec[e + f*x]] + Sec[e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan [e + f*x]^2]] + Sqrt[2]*(Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqr t[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]] - Log [1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt [1 + Sec[e + f*x]]*Sqrt[Tan[e + f*x]^2]]))*Sqrt[Tan[e + f*x]^2]))/(c*f*(-1 + Cos[e + f*x])*(1 + Cos[e + f*x])^2*(-1 + Sec[e + f*x])*Sec[e + f*x]^(5/ 2)*Sqrt[a*(1 + Sec[e + f*x])])
Time = 0.53 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {3042, 4452, 27, 109, 27, 175, 65, 104, 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.
| \(\displaystyle \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4452 |
| \(\displaystyle -\frac {a c g \tan (e+f x) \int \frac {(g \sec (e+f x))^{3/2}}{a (\sec (e+f x)+1) (c-c \sec (e+f x))^{3/2}}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c g \tan (e+f x) \int \frac {(g \sec (e+f x))^{3/2}}{(\sec (e+f x)+1) (c-c \sec (e+f x))^{3/2}}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {\int \frac {c g^2 (2 \sec (e+f x)+1)}{2 \sqrt {g \sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)}{c^2}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \int \frac {2 \sec (e+f x)+1}{\sqrt {g \sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \left (2 \int \frac {1}{\sqrt {g \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)-\int \frac {1}{\sqrt {g \sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)\right )}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \left (4 \int \frac {1}{\frac {c \sec (e+f x) g}{c-c \sec (e+f x)}+g}d\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}-\int \frac {1}{\sqrt {g \sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)\right )}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \left (4 \int \frac {1}{\frac {c \sec (e+f x) g}{c-c \sec (e+f x)}+g}d\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}-2 \int \frac {1}{\frac {2 c \sec (e+f x) g}{c-c \sec (e+f x)}+g}d\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {c g \tan (e+f x) \left (\frac {g \sqrt {g \sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \left (\frac {4 \arctan \left (\frac {\sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} \sqrt {g}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} \sqrt {g}}\right )}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
-((c*g*(-1/2*(g^2*((4*ArcTan[(Sqrt[c]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[ c - c*Sec[e + f*x]])])/(Sqrt[c]*Sqrt[g]) - (Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[c ]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*Sqrt [g])))/c + (g*Sqrt[g*Sec[e + f*x]])/(c*Sqrt[c - c*Sec[e + f*x]]))*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]))
3.2.82.3.1 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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[a *c*g*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])) Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Cs c[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0]
Time = 5.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(-\frac {g^{2} \left (\operatorname {arcsinh}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ) \sqrt {2}\, \sin \left (f x +e \right )-2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )+2 \,\operatorname {arctanh}\left (\frac {-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\right ) \sin \left (f x +e \right )+2 \,\operatorname {arctanh}\left (\frac {\cos \left (f x +e \right )+\sin \left (f x +e \right )+1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\right ) \sin \left (f x +e \right )-2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {g \sec \left (f x +e \right )}\, \cot \left (f x +e \right )}{2 c f a \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(222\) |
-1/2/c/f/a*g^2*(arcsinh(cot(f*x+e)-csc(f*x+e))*2^(1/2)*sin(f*x+e)-2*(1/(co s(f*x+e)+1))^(1/2)*cos(f*x+e)+2*arctanh(1/2*(-cos(f*x+e)+sin(f*x+e)-1)/(co s(f*x+e)+1)/(1/(cos(f*x+e)+1))^(1/2))*sin(f*x+e)+2*arctanh(1/2*(cos(f*x+e) +sin(f*x+e)+1)/(cos(f*x+e)+1)/(1/(cos(f*x+e)+1))^(1/2))*sin(f*x+e)-2*(1/(c os(f*x+e)+1))^(1/2))*(a*(sec(f*x+e)+1))^(1/2)*(g*sec(f*x+e))^(1/2)/(cos(f* x+e)+1)/(1/(cos(f*x+e)+1))^(1/2)*cot(f*x+e)
Time = 0.38 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.18 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\left [\frac {\sqrt {2} a g^{2} \sqrt {\frac {g}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - g \cos \left (f x + e\right )^{2} + 2 \, g \cos \left (f x + e\right ) + 3 \, g}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt {\frac {g}{a}} \log \left (\frac {g \cos \left (f x + e\right )^{3} + 4 \, {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, g \cos \left (f x + e\right )^{2} + 8 \, g}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, g^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{4 \, a c f \sin \left (f x + e\right )}, -\frac {\sqrt {2} a g^{2} \sqrt {-\frac {g}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{g \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt {-\frac {g}{a}} \arctan \left (\frac {2 \, \sqrt {-\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{g \cos \left (f x + e\right )^{2} - g \cos \left (f x + e\right ) - 2 \, g}\right ) \sin \left (f x + e\right ) - 2 \, g^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, a c f \sin \left (f x + e\right )}\right ] \]
[1/4*(sqrt(2)*a*g^2*sqrt(g/a)*log((2*sqrt(2)*sqrt(g/a)*sqrt((a*cos(f*x + e ) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - g*co s(f*x + e)^2 + 2*g*cos(f*x + e) + 3*g)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1))*sin(f*x + e) + 2*a*g^2*sqrt(g/a)*log((g*cos(f*x + e)^3 + 4*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt(g/a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*s qrt(g/cos(f*x + e))*sin(f*x + e) - 7*g*cos(f*x + e)^2 + 8*g)/(cos(f*x + e) ^3 + cos(f*x + e)^2))*sin(f*x + e) + 4*g^2*sqrt((a*cos(f*x + e) + a)/cos(f *x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e))/(a*c*f*sin(f*x + e)), -1/2*(sq rt(2)*a*g^2*sqrt(-g/a)*arctan(sqrt(2)*sqrt(-g/a)*sqrt((a*cos(f*x + e) + a) /cos(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e)/(g*sin(f*x + e)))*sin(f*x + e) + 2*a*g^2*sqrt(-g/a)*arctan(2*sqrt(-g/a)*sqrt((a*cos(f*x + e) + a)/c os(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e)*sin(f*x + e)/(g*cos(f*x + e )^2 - g*cos(f*x + e) - 2*g))*sin(f*x + e) - 2*g^2*sqrt((a*cos(f*x + e) + a )/cos(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e))/(a*c*f*sin(f*x + e))]
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1400 vs. \(2 (149) = 298\).
Time = 0.45 (sec) , antiderivative size = 1400, normalized size of antiderivative = 7.82 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\text {Too large to display} \]
1/2*(4*g^2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))*sin(1/2*ar ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*g^2*cos(1/2*arctan2(sin(2*f *x + 2*e), cos(2*f*x + 2*e)))*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*g^2*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (s qrt(2)*g^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2 )*g^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*g ^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(2)*g^2)*log (2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/4*arct an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) + (sqrt(2)*g^2*cos(1/2*arctan2(sin(2*f*x + 2 *e), cos(2*f*x + 2*e)))^2 + sqrt(2)*g^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*g^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos (2*f*x + 2*e))) + sqrt(2)*g^2)*log(2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos (2*f*x + 2*e)))^2 + 2*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) ^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*sq rt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) - (sqrt(2) *g^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*g^2* sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*g^2*cos (1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(2)*g^2)*log(2*...
\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\int { -\frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) - c\right )}} \,d x } \]
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]