\(\int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx\) [242]

Optimal result

Integrand size = 39, antiderivative size = 167 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=-\frac {\sqrt {2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) f}+\frac {2 \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) \sqrt {c+d} f} \]

[Out]

-g^(3/2)*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*g^(3/2)*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^(1/2)/(c-d)/f/a^(1/2)/(c+d)^(1/2)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4059, 3893, 214, 4050} \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} f (c-d)} \]

[In]

Int[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]

[Out]

-((Sqrt[2]*g^(3/2)*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*Sqrt[c]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sq
rt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - 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 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 4059

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))), x_Symbol] :> Dist[(-a)*(g/(b*c - a*d)), Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x]
, x] + Dist[c*(g/(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
] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

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

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.19 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {g \cos \left (\frac {1}{2} (e+f x)\right ) \left (2 \sqrt {c+d} \log \left (\cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )\right )-2 \sqrt {c+d} \log \left (\cos \left (\frac {1}{4} (e+f x)\right )+\sin \left (\frac {1}{4} (e+f x)\right )\right )+\sqrt {2} \sqrt {c} \left (-\log \left (\sqrt {2} \sqrt {c+d}-2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sqrt {2} \sqrt {c+d}+2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) \sqrt {g \sec (e+f x)}}{(c-d) \sqrt {c+d} f \sqrt {a (1+\sec (e+f x))}} \]

[In]

Integrate[(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])),x]

[Out]

(g*Cos[(e + f*x)/2]*(2*Sqrt[c + d]*Log[Cos[(e + f*x)/4] - Sin[(e + f*x)/4]] - 2*Sqrt[c + d]*Log[Cos[(e + f*x)/
4] + Sin[(e + f*x)/4]] + Sqrt[2]*Sqrt[c]*(-Log[Sqrt[2]*Sqrt[c + d] - 2*Sqrt[c]*Sin[(e + f*x)/2]] + Log[Sqrt[2]
*Sqrt[c + d] + 2*Sqrt[c]*Sin[(e + f*x)/2]]))*Sqrt[g*Sec[e + f*x]])/((c - 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. \(556\) vs. \(2(132)=264\).

Time = 20.66 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.34

method	result	size
default	\(\frac {g \sqrt {-\frac {g \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (c \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d -2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+2 c -2 d}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )-c \sqrt {2}\, \ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+c -d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )-2 \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}}\right )}{2 f a \sqrt {\frac {c}{c -d}}\, \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\)	\(557\)

[In]

int((g*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*g/f/a/(c/(c-d))^(1/2)/(c-d)/((c+d)*(c-d))^(1/2)*(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)/((1-cos(f*x+e))^2*cs
c(f*x+e)^2-1))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*(c*2^(1/
2)*ln(2*(2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*c-2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e
)^2+1)^(1/2)*(c/(c-d))^(1/2)*d-((c+d)*(c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e))+c-d)/(c*(-cot(f*x+e)+csc(f*x+e))-(
-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d))^(1/2)))-c*2^(1/2)*ln(-2*(2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/
2)*(c/(c-d))^(1/2)*c-2^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)*(c/(c-d))^(1/2)*d+((c+d)*(c-d))^(1/2)*(-c
ot(f*x+e)+csc(f*x+e))+c-d)/(-c*(-cot(f*x+e)+csc(f*x+e))+(-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d))^(1/2)))-2*((c
+d)*(c-d))^(1/2)*arcsinh(cot(f*x+e)-csc(f*x+e))*(c/(c-d))^(1/2))/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.80 (sec) , antiderivative size = 1103, normalized size of antiderivative = 6.60 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]

[In]

integrate((g*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(2)*g*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
)) + sqrt(c*g/(a*c + a*d))*g*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)*f), 1/2*(2*sqrt(2)*g*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))) - sqrt(c*g/(a*c + a*d))*g*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)*f), -1/2*(sqrt(2)*g
*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*sqrt(-c*g
/(a*c + a*d))*g*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)*f), (sqrt(2)*g*sqrt(-g/a)*arctan(sq
rt(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))) +
sqrt(-c*g/(a*c + a*d))*g*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)*f)]

Sympy [F]

\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (c + d \sec {\left (e + f x \right )}\right )}\, dx \]

[In]

integrate((g*sec(f*x+e))**(3/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))**(1/2),x)

[Out]

Integral((g*sec(e + f*x))**(3/2)/(sqrt(a*(sec(e + f*x) + 1))*(c + d*sec(e + f*x))), x)

Maxima [F]

\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]

[In]

integrate((g*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

1/2*(sqrt(2)*c*f*g*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*cos(f*x + e) + c^2)*sin(2*f*x + 2*e) + (c^2 - 2*c*d)*sin(f*x + e))*sin(1/2*a
rctan2(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*co
s(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*f*g*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*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), c
os(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)*d*g*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*g*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(g)/((c*d - d^2)*sqrt(a)*f)

Giac [F(-2)]

Exception generated. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((g*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]

[In]

int((g/cos(e + f*x))^(3/2)/((a + a/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))),x)

[Out]

int((g/cos(e + f*x))^(3/2)/((a + a/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))), x)