Integrand size = 38, antiderivative size = 140 \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c f}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {\sec (e+f x)} \sin (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {a} c f}+\frac {\csc (e+f x) \sqrt {a+a \sec (e+f x)}}{a c f \sqrt {\sec (e+f x)}} \]
-2*arcsinh(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/c/f/a^(1/2)+1/2*arct anh(1/2*sin(f*x+e)*a^(1/2)*sec(f*x+e)^(1/2)*2^(1/2)/(a+a*sec(f*x+e))^(1/2) )/c/f*2^(1/2)/a^(1/2)+csc(f*x+e)*(a+a*sec(f*x+e))^(1/2)/a/c/f/sec(f*x+e)^( 1/2)
Leaf count is larger than twice the leaf count of optimal. \(724\) vs. \(2(140)=280\).
Time = 11.25 (sec) , antiderivative size = 724, normalized size of antiderivative = 5.17 \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\frac {\sec ^{\frac {3}{2}}(e+f x) \sqrt {(1+\cos (e+f x)) \sec (e+f x)} \sqrt {1+\sec (e+f x)} \left (-\frac {2 \cot (e)}{f}+\frac {\csc \left (\frac {e}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}\right ) \sin \left (\frac {f x}{2}\right )}{f}+\frac {\sec \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}+\frac {f x}{2}\right ) \sin \left (\frac {f x}{2}\right )}{f}\right ) \sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{\sqrt {a (1+\sec (e+f x))} (c-c \sec (e+f x))}+\frac {\cos (e+f x) \left (\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)-2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {-1+\sec ^2(e+f x)}\right )-\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)+2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {-1+\sec ^2(e+f x)}\right )\right ) (1+\sec (e+f x))^{3/2} \sqrt {-1+\sec ^2(e+f x)} \sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sin (e+f x)}{2 f (1+\cos (e+f x)) \sqrt {2-2 \cos ^2(e+f x)} \sqrt {1-\cos ^2(e+f x)} \sqrt {a (1+\sec (e+f x))} (c-c \sec (e+f x))}+\frac {\cos (e+f x) \left (-8 \log (1+\sec (e+f x))+8 \log \left (\sqrt {\sec (e+f x)}+\sec ^{\frac {3}{2}}(e+f x)+\sqrt {1+\sec (e+f x)} \sqrt {-1+\sec ^2(e+f x)}\right )+\sqrt {2} \left (-\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)-2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {-1+\sec ^2(e+f x)}\right )+\log \left (1-2 \sec (e+f x)-3 \sec ^2(e+f x)+2 \sqrt {2} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)} \sqrt {-1+\sec ^2(e+f x)}\right )\right )\right ) (1+\sec (e+f x))^{3/2} \sqrt {-1+\sec ^2(e+f x)} \sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sin (e+f x)}{2 f (1+\cos (e+f x)) \left (1-\cos ^2(e+f x)\right ) \sqrt {a (1+\sec (e+f x))} (c-c \sec (e+f x))} \]
(Sec[e + f*x]^(3/2)*Sqrt[(1 + Cos[e + f*x])*Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*((-2*Cot[e])/f + (Csc[e/2]*Csc[e/2 + (f*x)/2]*Sin[(f*x)/2])/f + (Se c[e/2]*Sec[e/2 + (f*x)/2]*Sin[(f*x)/2])/f)*Sin[e/2 + (f*x)/2]^2)/(Sqrt[a*( 1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(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[-1 + Sec[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[-1 + Sec[e + f*x]^2]])*(1 + Sec[e + f*x])^(3/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f *x)/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*Sqrt[2 - 2*Cos[e + f*x]^2]* Sqrt[1 - Cos[e + f*x]^2]*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(-8*Log[1 + Sec[e + f*x]] + 8*Log[Sqrt[Sec[e + f*x]] + Sec [e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]] + Sqrt [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[-1 + Sec[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[-1 + Sec[e + f*x]^2]]))*(1 + Sec[e + f*x])^(3/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f*x)/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*(1 - Cos[e + f*x]^2)*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x]))
Time = 0.46 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 4452, 27, 109, 27, 175, 64, 104, 216, 223}
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.
| \(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )^{5/2}}{\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4452 |
| \(\displaystyle -\frac {a c \tan (e+f x) \int \frac {\sec ^{\frac {3}{2}}(e+f x)}{a (\sec (e+f x)+1) (c-c \sec (e+f x))^{3/2}}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \tan (e+f x) \int \frac {\sec ^{\frac {3}{2}}(e+f x)}{(\sec (e+f x)+1) (c-c \sec (e+f x))^{3/2}}d\sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {\int \frac {c (2 \sec (e+f x)+1)}{2 \sqrt {\sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)}{c^2}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {\int \frac {2 \sec (e+f x)+1}{\sqrt {\sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {2 \int \frac {1}{\sqrt {\sec (e+f x)} \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)-\int \frac {1}{\sqrt {\sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {-\int \frac {1}{\sqrt {\sec (e+f x)} (\sec (e+f x)+1) \sqrt {c-c \sec (e+f x)}}d\sec (e+f x)-\frac {4 \int \frac {1}{\sqrt {1-\frac {c-c \sec (e+f x)}{c}}}d\sqrt {c-c \sec (e+f x)}}{c}}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {-2 \int \frac {1}{\frac {2 c \sec (e+f x)}{c-c \sec (e+f x)}+1}d\frac {\sqrt {\sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}-\frac {4 \int \frac {1}{\sqrt {1-\frac {c-c \sec (e+f x)}{c}}}d\sqrt {c-c \sec (e+f x)}}{c}}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {-\frac {4 \int \frac {1}{\sqrt {1-\frac {c-c \sec (e+f x)}{c}}}d\sqrt {c-c \sec (e+f x)}}{c}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {\sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c}}}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c \tan (e+f x) \left (\frac {\sqrt {\sec (e+f x)}}{c \sqrt {c-c \sec (e+f x)}}-\frac {-\frac {4 \arcsin \left (\frac {\sqrt {c-c \sec (e+f x)}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {\sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c}}}{2 c}\right )}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}\) |
-((c*(-1/2*((-4*ArcSin[Sqrt[c - c*Sec[e + f*x]]/Sqrt[c]])/Sqrt[c] - (Sqrt[ 2]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]]])/ Sqrt[c])/c + Sqrt[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]]))
3.2.80.3.1 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[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
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[(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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(120)=240\).
Time = 5.05 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(-\frac {\left (\sqrt {2}\, \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {2}}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}}\right ) \sin \left (f x +e \right )-2 \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )-2 \arctan \left (\frac {-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}}\right ) \sin \left (f x +e \right )-2 \arctan \left (\frac {\cos \left (f x +e \right )+\sin \left (f x +e \right )+1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}}\right ) \sin \left (f x +e \right )-2 \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sec \left (f x +e \right )^{\frac {5}{2}} \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )}{2 c f a \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(255\) |
-1/2/c/f/a*(2^(1/2)*arctan(1/2*sin(f*x+e)*2^(1/2)/(cos(f*x+e)+1)/(-1/(cos( f*x+e)+1))^(1/2))*sin(f*x+e)-2*(-1/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)-2*arct an(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*arctan(1/2*(cos(f*x+e)+sin(f*x+e)+1)/(cos(f*x+e)+1)/(-1/(co s(f*x+e)+1))^(1/2))*sin(f*x+e)-2*(-1/(cos(f*x+e)+1))^(1/2))*(a*(sec(f*x+e) +1))^(1/2)*sec(f*x+e)^(5/2)/(cos(f*x+e)+1)/(-1/(cos(f*x+e)+1))^(1/2)*cos(f *x+e)^2*cot(f*x+e)
Time = 0.31 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.30 \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\left [\frac {\sqrt {2} \sqrt {a} \log \left (-\frac {\cos \left (f x + e\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )} \sin \left (f x + e\right )}{\sqrt {a}} - 2 \, \cos \left (f x + e\right ) - 3}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + \frac {4 \, {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{\sqrt {\cos \left (f x + e\right )}} + 8 \, a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )}}{4 \, a c f \sin \left (f x + e\right )}, -\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \sqrt {\cos \left (f x + e\right )}}{\sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )} \sin \left (f x + e\right )}{a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a}\right ) \sin \left (f x + e\right ) - 2 \, \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )}}{2 \, a c f \sin \left (f x + e\right )}\right ] \]
[1/4*(sqrt(2)*sqrt(a)*log(-(cos(f*x + e)^2 - 2*sqrt(2)*sqrt((a*cos(f*x + e ) + a)/cos(f*x + e))*sqrt(cos(f*x + e))*sin(f*x + e)/sqrt(a) - 2*cos(f*x + e) - 3)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1))*sin(f*x + e) + 2*sqrt(a)*l og((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e)/sqrt(co s(f*x + e)) + 8*a)/(cos(f*x + e)^3 + cos(f*x + e)^2))*sin(f*x + e) + 4*sqr t((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x + e)))/(a*c*f*sin(f*x + e)), -1/2*(sqrt(2)*a*sqrt(-1/a)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/c os(f*x + e))*sqrt(-1/a)*sqrt(cos(f*x + e))/sin(f*x + e))*sin(f*x + e) + 2* sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(co s(f*x + e))*sin(f*x + e)/(a*cos(f*x + e)^2 - a*cos(f*x + e) - 2*a))*sin(f* x + e) - 2*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x + e)))/(a* c*f*sin(f*x + e))]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1310 vs. \(2 (120) = 240\).
Time = 0.45 (sec) , antiderivative size = 1310, normalized size of antiderivative = 9.36 \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\text {Too large to display} \]
-1/2*((sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sq rt(2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*c os(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(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*sqrt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e))) + 2) - (sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f *x + 2*e)))^2 + sqrt(2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) ))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sq rt(2))*log(2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*si n(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sqrt(2)*cos(1/4*a rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*sqrt(2)*sin(1/4*arctan2(si n(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) + (sqrt(2)*cos(1/2*arctan2(sin(2*f *x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*sin(1/2*arctan2(sin(2*f*x + 2*e) , cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) + sqrt(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*sqrt(2)* sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) - (sqrt(2)*cos(1 /2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*sin(1/2*arc...
\[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\int { -\frac {\sec \left (f x + e\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) - c\right )}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx=\int \frac {{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]