\(\int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx\) [243]

Optimal result

Integrand size = 39, antiderivative size = 231 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} d f}+\frac {\sqrt {2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) f}-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) d \sqrt {c+d} f} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4063, 4050, 214, 4108, 3893, 3887} \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=-\frac {2 c^{3/2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f (c-d) \sqrt {c+d}}+\frac {\sqrt {2} g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} f (c-d)}+\frac {2 g^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} d f} \]

[In]

Int[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]

[Out]

(2*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*d
*f) + (Sqrt[2]*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e +
f*x]])])/(Sqrt[a]*(c - d)*f) - (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)

Rule 214

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]

Rule 3887

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-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] /; F
reeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] &&  !GtQ[a*(d/b), 0]

Rule 3893

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-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]

Rule 4050

Int[(Sqrt[csc[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*
(d_.) + (c_)), x_Symbol] :> Dist[-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]

Rule 4063

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(5/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))), x_Symbol] :> Dist[(-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] + Dist[g^2/(d*(b*c - a*d)), Int[Sqrt[g*Csc[e + f*x]]*((a*c + (b*c - a*d)*C
sc[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]

Rule 4108

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Dist[(A*b - a*B)/b, Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {g^2 \int \frac {\sqrt {g \sec (e+f x)} (a c+(a c-a d) \sec (e+f x))}{\sqrt {a+a \sec (e+f x)}} \, dx}{a (c-d) d}-\frac {\left (c^2 g^2\right ) \int \frac {\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx}{a (c-d) d} \\ & = \frac {g^2 \int \frac {\sqrt {g \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx}{c-d}+\frac {g^2 \int \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)} \, dx}{a d}+\frac {\left (2 c^2 g^3\right ) \text {Subst}\left (\int \frac {1}{a c+a d-c g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{(c-d) 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}-\frac {\left (2 g^3\right ) \text {Subst}\left (\int \frac {1}{2 a-g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{(c-d) f}-\frac {\left (2 g^3\right ) \text {Subst}\left (\int \frac {1}{a-g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.67 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g^2 \left (d \sqrt {c+d} \text {arctanh}\left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {2} \left ((c-d) \sqrt {c+d} \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )-c^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {g \sec (e+f x)}}{(c-d) d \sqrt {c+d} f \sqrt {a (1+\sec (e+f x))}} \]

[In]

Integrate[(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]

[Out]

(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])])

Maple [B] (warning: unable to verify)

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}}}\)	\(796\)

[In]

int((g*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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+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-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/(cos(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)

Fricas [A] (verification not implemented)

none

Time = 52.75 (sec) , antiderivative size = 1597, normalized size of antiderivative = 6.91 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]

[In]

integrate((g*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[-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*co
s(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*(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)) + 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*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*(2*sqrt(2)*d*g^2*sqrt(-g/a)*arctan(sqrt(2)*sqrt(-g/a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqr
t(g/cos(f*x + e))*cos(f*x + e)/(g*sin(f*x + e))) - 2*(c - d)*g^2*sqrt(-g/a)*arctan(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 - g*cos(f*x + e) -
2*g)) + 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 - d^2)*f), -(sqrt(2)*d*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))) - (c - d)*g^2*sqrt(-g/a)*arctan(2*sqrt(-g/a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqr
t(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)) + 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 - d^2)*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Timed out} \]

[In]

integrate((g*sec(f*x+e))**(5/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]

[In]

integrate((g*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

-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)*c
os(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*cos(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*arctan2(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)*si
n(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*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)*co
s(2*f*x + 2*e))*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2), x) - sqrt(2)*d^2*g^2*log(cos(1/2*arctan2(sin(
f*x + e), cos(f*x + e)))^2 + sin(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e),
cos(f*x + e))) + 1) + sqrt(2)*d^2*g^2*log(cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + sin(1/2*arctan2(sin
(f*x + e), cos(f*x + e)))^2 - 2*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 1) - (c*d - d^2)*g^2*log(2*cos(
1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sqrt(2)*cos(
1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2*sqrt(2)*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2) + (c*d
- d^2)*g^2*log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e)
))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) - 2*sqrt(2)*sin(1/2*arctan2(sin(f*x + e), cos(f*
x + e))) + 2) - (c*d - d^2)*g^2*log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*sin(1/2*arctan2(sin(f
*x + e), cos(f*x + e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2*sqrt(2)*sin(1/2*arctan2
(sin(f*x + e), cos(f*x + e))) + 2) + (c*d - d^2)*g^2*log(2*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 + 2*
sin(1/2*arctan2(sin(f*x + e), cos(f*x + e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(f*x + e), cos(f*x + e))) - 2*sq
rt(2)*sin(1/2*arctan2(sin(f*x + e), cos(f*x + e))) + 2))*sqrt(g)/((c*d^2 - d^3)*sqrt(a)*f)

Giac [F(-2)]

Exception generated. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:index.cc index_m
i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]

[In]

int((g/cos(e + f*x))^(5/2)/((a + a/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))),x)

[Out]

int((g/cos(e + f*x))^(5/2)/((a + a/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))), x)