\(\int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx\) [271]

Optimal result

Integrand size = 39, antiderivative size = 83 \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}} \]

[Out]

2*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*a/(a+b),2^(1/2)*(c/(c+d))^
(1/2))*((d+c*cos(f*x+e))/(c+d))^(1/2)*(g*sec(f*x+e))^(1/2)/(a+b)/f/(c+d*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4060, 2886, 2884} \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)}} \]

[In]

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

[Out]

(2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (e + f*x)/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f
*x]])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]])

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 4060

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))), x_Symbol] :> Dist[g*Sqrt[g*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]),
Int[1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (g \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(b+a \cos (e+f x)) \sqrt {d+c \cos (e+f x)}} \, dx}{\sqrt {c+d \sec (e+f x)}} \\ & = \frac {\left (g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(b+a \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{\sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}} \]

[In]

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

[Out]

(2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (e + f*x)/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f
*x]])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.34 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69

method	result	size
default	\(-\frac {2 i g \cos \left (f x +e \right ) \sqrt {c +d \sec \left (f x +e \right )}\, \left (2 a \operatorname {EllipticPi}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -\frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right )-a \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right )-b \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right )\right ) \sqrt {g \sec \left (f x +e \right )}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f \left (a -b \right ) \left (a +b \right ) \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\)	\(223\)

[In]

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

[Out]

-2*I*g/f/(a-b)/(a+b)*cos(f*x+e)*(c+d*sec(f*x+e))^(1/2)*(2*a*EllipticPi(I*(-cot(f*x+e)+csc(f*x+e)),-(a-b)/(a+b)
,I*((c-d)/(c+d))^(1/2))-a*EllipticF(I*(-cot(f*x+e)+csc(f*x+e)),(-(c-d)/(c+d))^(1/2))-b*EllipticF(I*(-cot(f*x+e
)+csc(f*x+e)),(-(c-d)/(c+d))^(1/2)))*(g*sec(f*x+e))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)/(d+c
*cos(f*x+e))/(1/(cos(f*x+e)+1))^(1/2)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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