3.3.84 \(\int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx\) [284]

Optimal. Leaf size=166 \[ \frac {2 g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.60, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4064, 3944, 2886, 2884, 4060} \begin {gather*} \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{d f (c+d) \sqrt {a+b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*f
*Sqrt[a + b*Sec[e + f*x]]) - (2*c*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), (e + f*x)/2
, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*(c + d)*f*Sqrt[a + b*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 3944

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

Rule 4064

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(5/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))), x_Symbol] :> Dist[g/d, Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[c*(
g/d), Int[(g*Csc[e + f*x])^(3/2)/(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] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx &=\frac {g \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx}{d}-\frac {(c g) \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx}{d}\\ &=\frac {\left (g^2 \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {b+a \cos (e+f x)}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left (c g^2 \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))} \, dx}{d \sqrt {a+b \sec (e+f x)}}\\ &=\frac {\left (g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left (c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}} (d+c \cos (e+f x))} \, dx}{d \sqrt {a+b \sec (e+f x)}}\\ &=\frac {2 g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.68, size = 246, normalized size = 1.48 \begin {gather*} -\frac {2 i g \sqrt {-\frac {a (-1+\cos (e+f x))}{a+b}} \sqrt {\frac {a (1+\cos (e+f x))}{a-b}} \sqrt {b+a \cos (e+f x)} \cot (e+f x) \left ((-b c+a d) \Pi \left (1-\frac {a}{b};i \sinh ^{-1}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right )|\frac {-a+b}{a+b}\right )+b c \Pi \left (\frac {(a-b) c}{-b c+a d};i \sinh ^{-1}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right )|\frac {-a+b}{a+b}\right )\right ) (g \sec (e+f x))^{3/2}}{\sqrt {\frac {1}{a-b}} b d (-b c+a d) f \sqrt {a+b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*g*Sqrt[-((a*(-1 + Cos[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Cos[e + f*x]))/(a - b)]*Sqrt[b + a*Cos[e + f*x
]]*Cot[e + f*x]*((-(b*c) + a*d)*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f*x]]], (-
a + b)/(a + b)] + b*c*EllipticPi[((a - b)*c)/(-(b*c) + a*d), I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[e + f
*x]]], (-a + b)/(a + b)])*(g*Sec[e + f*x])^(3/2))/(Sqrt[(a - b)^(-1)]*b*d*(-(b*c) + a*d)*f*Sqrt[a + b*Sec[e +
f*x]])

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Maple [C] Result contains complex when optimal does not.
time = 6.28, size = 346, normalized size = 2.08

method	result	size
default	\(\frac {2 i \left (\EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right ) d c +\EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right ) d^{2}+2 c^{2} \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -1, i \sqrt {\frac {a -b}{a +b}}\right )-2 \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -1, i \sqrt {\frac {a -b}{a +b}}\right ) d^{2}-2 c^{2} \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -\frac {c -d}{c +d}, i \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\cos ^{3}\left (f x +e \right )\right )}{f \left (a \cos \left (f x +e \right )+b \right ) \left (\frac {1}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{4} d \left (c +d \right ) \left (c -d \right )}\)	\(346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I/f*(EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b))^(1/2))*d*c+EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e)
,(-(a-b)/(a+b))^(1/2))*d^2+2*c^2*EllipticPi(I*(-1+cos(f*x+e))/sin(f*x+e),-1,I*((a-b)/(a+b))^(1/2))-2*EllipticP
i(I*(-1+cos(f*x+e))/sin(f*x+e),-1,I*((a-b)/(a+b))^(1/2))*d^2-2*c^2*EllipticPi(I*(-1+cos(f*x+e))/sin(f*x+e),-(c
-d)/(c+d),I*((a-b)/(a+b))^(1/2)))*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((a*cos(f*x+e)+b)/cos(f*x+e))^
(1/2)*(-1+cos(f*x+e))^2*(g/cos(f*x+e))^(5/2)*cos(f*x+e)^3/(a*cos(f*x+e)+b)/(1/(cos(f*x+e)+1))^(5/2)/sin(f*x+e)
^4/d/(c+d)/(c-d)

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

[Out]

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

[Out]

Timed out

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3007 deep

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

[Out]

integrate((g*sec(f*x + e))^(5/2)/(sqrt(b*sec(f*x + e) + a)*(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 )}^{5/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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