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

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

[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)

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Rubi [A]
time = 0.28, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 = {4060, 2886, 2884} \begin {gather*} \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)}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.25, size = 83, normalized size = 1.00 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 6.54, size = 254, normalized size = 3.06

method	result	size
default	\(-\frac {2 i \left (2 a \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -\frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right )-a \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {c -d}{c +d}}\right )-b \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {c -d}{c +d}}\right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \left (c +d \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right ) \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right )}{f \left (d +c \cos \left (f x +e \right )\right ) \left (\frac {1}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{2} \left (a -b \right ) \left (a +b \right )}\)	\(254\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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