Integrand size = 39, antiderivative size = 231 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} d f}+\frac {\sqrt {2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) f}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) d \sqrt {c+d} f} \]
2*g^(5/2)*arctanh(a^(1/2)*g^(1/2)*tan(f*x+e)/(g*sec(f*x+e))^(1/2)/(a+a*sec (f*x+e))^(1/2))/d/f/a^(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))*2^(1/2)/(c-d)/f/a^(1 /2)-2*c^(3/2)*g^(5/2)*arctanh(a^(1/2)*c^(1/2)*g^(1/2)*tan(f*x+e)/(c+d)^(1/ 2)/(g*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2))/(c-d)/d/f/a^(1/2)/(c+d)^(1 /2)
Time = 1.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.67 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g^2 \left (d \sqrt {c+d} \text {arctanh}\left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {2} \left ((c-d) \sqrt {c+d} \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )-c^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {g \sec (e+f x)}}{(c-d) d \sqrt {c+d} f \sqrt {a (1+\sec (e+f x))}} \]
(2*g^2*(d*Sqrt[c + d]*ArcTanh[Sin[(e + f*x)/2]] + Sqrt[2]*((c - d)*Sqrt[c + d]*ArcTanh[Sqrt[2]*Sin[(e + f*x)/2]] - c^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[c]* Sin[(e + f*x)/2])/Sqrt[c + d]]))*Cos[(e + f*x)/2]*Sqrt[g*Sec[e + f*x]])/(( c - d)*d*Sqrt[c + d]*f*Sqrt[a*(1 + Sec[e + f*x])])
Time = 1.41 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3042, 4466, 3042, 4453, 221, 4511, 3042, 4289, 221, 4295, 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.
| \(\displaystyle \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a \sec (e+f x)+a} (c+d \sec (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4466 |
| \(\displaystyle \frac {g^2 \int \frac {\sqrt {g \sec (e+f x)} (a c+a (c-d) \sec (e+f x))}{\sqrt {\sec (e+f x) a+a}}dx}{a d (c-d)}-\frac {c^2 g^2 \int \frac {\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}{c+d \sec (e+f x)}dx}{a d (c-d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \left (a c+a (c-d) \csc \left (e+f x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a d (c-d)}-\frac {c^2 g^2 \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a d (c-d)}\) |
| \(\Big \downarrow \) 4453 |
| \(\displaystyle \frac {2 c^2 g^3 \int \frac {1}{a (c+d)-\frac {a^2 c \sin (e+f x) \tan (e+f x)}{\sec (e+f x) a+a}}d\left (-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}\right )}{d f (c-d)}+\frac {g^2 \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \left (a c+a (c-d) \csc \left (e+f x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a d (c-d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {g^2 \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \left (a c+a (c-d) \csc \left (e+f x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 4511 |
| \(\displaystyle \frac {g^2 \left ((c-d) \int \sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}dx+a d \int \frac {\sqrt {g \sec (e+f x)}}{\sqrt {\sec (e+f x) a+a}}dx\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left ((c-d) \int \sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx+a d \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 4289 |
| \(\displaystyle \frac {g^2 \left (a d \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx-\frac {2 a g (c-d) \int \frac {1}{a-\frac {a^2 \sin (e+f x) \tan (e+f x)}{\sec (e+f x) a+a}}d\left (-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}\right )}{f}\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {g^2 \left (a d \int \frac {\sqrt {g \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx+\frac {2 \sqrt {a} \sqrt {g} (c-d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{f}\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 4295 |
| \(\displaystyle \frac {g^2 \left (\frac {2 \sqrt {a} \sqrt {g} (c-d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{f}-\frac {2 a d g \int \frac {1}{2 a-\frac {a^2 \sin (e+f x) \tan (e+f x)}{\sec (e+f x) a+a}}d\left (-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}\right )}{f}\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {g^2 \left (\frac {2 \sqrt {a} \sqrt {g} (c-d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{f}+\frac {\sqrt {2} \sqrt {a} d \sqrt {g} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{f}\right )}{a d (c-d)}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}\) |
(g^2*((2*Sqrt[a]*(c - d)*Sqrt[g]*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(S qrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/f + (Sqrt[2]*Sqrt[a]*d*Sqr t[g]*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]* Sqrt[a + a*Sec[e + f*x]])])/f))/(a*(c - d)*d) - (2*c^(3/2)*g^(5/2)*ArcTanh [(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sec[e + f*x]]* Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*d*Sqrt[c + d]*f)
3.3.43.3.1 Defintions of rubi rules used
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[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(d/f) Subst[Int[1/(b - d*x^2), x], x, b*( Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]))], x] /; FreeQ [{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && !GtQ[a*(d/b), 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(d/(a*f)) Subst[Int[1/(2*b - d*x^2), x], x, b*(Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]))], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(Sqrt[csc[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[-2*b*(g /f) Subst[Int[1/(b*c + a*d - c*g*x^2), x], x, b*(Cot[e + f*x]/(Sqrt[g*Csc [e + f*x]]*Sqrt[a + b*Csc[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(5/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[(-c^2 )*(g^2/(d*(b*c - a*d))) Int[Sqrt[g*Csc[e + f*x]]*(Sqrt[a + b*Csc[e + f*x] ]/(c + d*Csc[e + f*x])), x], x] + Simp[g^2/(d*(b*c - a*d)) Int[Sqrt[g*Csc [e + f*x]]*((a*c + (b*c - a*d)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^ 2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)/b Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Simp[B/b Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b , d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(795\) vs. \(2(184)=368\).
Time = 22.87 (sec) , antiderivative size = 796, normalized size of antiderivative = 3.45
| method | result | size |
| default | \(-\frac {2 \left (\sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arcsinh}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ) \sqrt {\frac {c}{c -d}}\, d \sqrt {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 ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, c +\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 ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, d -\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 ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, c +\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 ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, d +\ln \left (-\frac {2 \left (2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {c}{c -d}}\, c \sin \left (f x +e \right )-2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {c}{c -d}}\, d \sin \left (f x +e \right )+\sin \left (f x +e \right ) c -\sin \left (f x +e \right ) d -\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )+c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d}\right ) c^{2}-\ln \left (-\frac {2 \left (-2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, c \cos \left (f x +e \right )+2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) d -2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {c}{c -d}}\, c +2 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {c}{c -d}}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \cos \left (f x +e \right )-c +d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+\sin \left (f x +e \right ) c -\sin \left (f x +e \right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right ) c^{2}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {g \sec \left (f x +e \right )}\, g^{2} \cos \left (f x +e \right )}{f a \sqrt {\frac {c}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-c +d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \left (c -d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(796\) |
-2/f/a/(c/(c-d))^(1/2)/((c+d)*(c-d))^(1/2)/(-c+d+((c+d)*(c-d))^(1/2))/(c-d +((c+d)*(c-d))^(1/2))*(((c+d)*(c-d))^(1/2)*arcsinh(cot(f*x+e)-csc(f*x+e))* (c/(c-d))^(1/2)*d*2^(1/2)-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))*(c/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*c+arcta nh(1/2*(cos(f*x+e)+sin(f*x+e)+1)/(cos(f*x+e)+1)/(1/(cos(f*x+e)+1))^(1/2))* (c/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*d-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))*(c/(c-d))^(1/2)*((c+d)*(c-d))^ (1/2)*c+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))*(c/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*d+ln(-2*(2*(1/(cos(f*x+e )+1))^(1/2)*(c/(c-d))^(1/2)*c*sin(f*x+e)-2*(1/(cos(f*x+e)+1))^(1/2)*(c/(c- d))^(1/2)*d*sin(f*x+e)+sin(f*x+e)*c-sin(f*x+e)*d-((c+d)*(c-d))^(1/2)*cos(f *x+e)+((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d* cos(f*x+e)-c+d))*c^2-ln(-2*(-2*(c/(c-d))^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*c* cos(f*x+e)+2*(c/(c-d))^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*d-2*(1/(c os(f*x+e)+1))^(1/2)*(c/(c-d))^(1/2)*c+2*(1/(cos(f*x+e)+1))^(1/2)*(c/(c-d)) ^(1/2)*d+((c+d)*(c-d))^(1/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)-c+d)/((( c+d)*(c-d))^(1/2)*cos(f*x+e)+sin(f*x+e)*c-sin(f*x+e)*d+((c+d)*(c-d))^(1/2) ))*c^2)*(a*(sec(f*x+e)+1))^(1/2)*(g*sec(f*x+e))^(1/2)*g^2/(cos(f*x+e)+1)/( 1/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)
Time = 52.75 (sec) , antiderivative size = 1597, normalized size of antiderivative = 6.91 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]
[-1/2*(sqrt(2)*d*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* cos(f*x + e)^2 - 2*g*cos(f*x + e) - 3*g)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + c*sqrt(c*g/(a*c + a*d))*g^2*log((c^2*g*cos(f*x + e)^3 - (7*c^2 + 6 *c*d)*g*cos(f*x + e)^2 - 4*((c^2 + c*d)*cos(f*x + e)^2 - (2*c^2 + 3*c*d + d^2)*cos(f*x + e))*sqrt(c*g/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(g/cos(f*x + e))*sin(f*x + e) + (2*c*d + d^2)*g*cos(f*x + e) + (8*c^2 + 8*c*d + d^2)*g)/(c^2*cos(f*x + e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^ 2 + d^2 + (2*c*d + d^2)*cos(f*x + e))) - (c - d)*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))*sqrt(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)))/((c*d - d^2)*f), -1/2*(sq rt(2)*d*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*cos(f*x + e)^2 - 2*g*cos(f*x + e) - 3*g)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 2 *c*sqrt(-c*g/(a*c + a*d))*g^2*arctan(1/2*(c*cos(f*x + e)^2 - (2*c + d)*cos (f*x + e))*sqrt(-c*g/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))* sqrt(g/cos(f*x + e))/(c*g*sin(f*x + e))) - (c - d)*g^2*sqrt(g/a)*log((g*co s(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))*sqrt(g/cos(f*x + e))*sin(f*x + e) - 7*g*cos(...
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]
-1/2*(sqrt(2)*c^2*f*g^2*integrate(((c^2*cos(2*f*x + 2*e)^2 + c^2*sin(2*f*x + 2*e)^2 - 2*(c*d - 2*d^2)*cos(f*x + e)^2 - (c^2 - 4*c*d)*sin(2*f*x + 2*e )*sin(f*x + e) - 2*(c*d - 2*d^2)*sin(f*x + e)^2 + (c^2 - (c^2 - 4*c*d)*cos (f*x + e))*cos(2*f*x + 2*e) - (c^2 - 2*c*d)*cos(f*x + e))*cos(1/2*arctan2( sin(f*x + e), cos(f*x + e))) - (c^2*cos(2*f*x + 2*e)*sin(f*x + e) - (c^2*c os(f*x + e) + c^2)*sin(2*f*x + 2*e) + (c^2 - 2*c*d)*sin(f*x + e))*sin(1/2* arctan2(sin(f*x + e), cos(f*x + e))))/((c^2*cos(2*f*x + 2*e)^2 + 4*d^2*cos (f*x + e)^2 + c^2*sin(2*f*x + 2*e)^2 + 4*c*d*sin(2*f*x + 2*e)*sin(f*x + e) + 4*d^2*sin(f*x + e)^2 + 4*c*d*cos(f*x + e) + c^2 + 2*(2*c*d*cos(f*x + e) + c^2)*cos(2*f*x + 2*e))*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + (c^2*cos(2*f*x + 2*e)^2 + 4*d^2*cos(f*x + e)^2 + c^2*sin(2*f*x + 2*e)^2 + 4*c*d*sin(2*f*x + 2*e)*sin(f*x + e) + 4*d^2*sin(f*x + e)^2 + 4*c*d*cos(f* x + e) + c^2 + 2*(2*c*d*cos(f*x + e) + c^2)*cos(2*f*x + 2*e))*sin(1/2*arct an2(sin(f*x + e), cos(f*x + e)))^2), x) + sqrt(2)*c^2*f*g^2*integrate(((2* c*d*cos(f*x + e)^2 + 2*c*d*sin(f*x + e)^2 - (c^2 - 2*c*d)*cos(2*f*x + 2*e) ^2 + c^2*cos(f*x + e) - (c^2 - 2*c*d)*sin(2*f*x + 2*e)^2 + (c^2 - 2*c*d + 4*d^2)*sin(2*f*x + 2*e)*sin(f*x + e) - (c^2 - 2*c*d - (c^2 - 2*c*d + 4*d^2 )*cos(f*x + e))*cos(2*f*x + 2*e))*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + (c^2*sin(f*x + e) + (c^2 + 2*c*d - 4*d^2)*cos(2*f*x + 2*e)*sin(f*x + e) - (c^2 - 2*c*d + (c^2 + 2*c*d - 4*d^2)*cos(f*x + e))*sin(2*f*x + 2...
Exception generated. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:index.cc index_m i_lex_is_greater Error: Bad Argument V alue
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]