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

Optimal. Leaf size=83 \[ \frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{f (a+b) \sqrt{c+d \sec (e+f x)}} \]

[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]])

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Rubi [A]  time = 0.433414, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3975, 2807, 2805} \[ \frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{f (a+b) \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[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 3975

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]

Rule 2807

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*
Sin[e + f*x])/(c + d)]), 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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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]

Rubi steps

\begin{align*} \int \frac{(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx &=\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}} \Pi \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]  time = 0.242946, size = 83, normalized size = 1. \[ \frac{2 g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{f (a+b) \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[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]])

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Maple [C]  time = 0.355, size = 236, normalized size = 2.8 \begin{align*}{\frac{-2\,i \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{f \left ( a+b \right ) \left ( a-b \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) } \left ( a{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ) +b{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ) -2\,a{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},i\sqrt{{\frac{c-d}{c+d}}} \right ) \right ) \sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}{\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I/f/(a+b)/(a-b)*(a*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),(-(c-d)/(c+d))^(1/2))+b*EllipticF(I*(-1+cos(f*x+e
))/sin(f*x+e),(-(c-d)/(c+d))^(1/2))-2*a*EllipticPi(I*(-1+cos(f*x+e))/sin(f*x+e),-(a-b)/(a+b),I*((c-d)/(c+d))^(
1/2)))*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(g/cos(f*x+e))^(3/2)*cos(f*x+e)^2*((d+c*cos(f*x+e))/cos
(f*x+e))^(1/2)/(d+c*cos(f*x+e))/(1/(1+cos(f*x+e)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)