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

Optimal. Leaf size=295 \[ \frac{g (a-b) \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (e+f x),\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}}-\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}}+\frac{2 b g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}} \]

[Out]

(g*(b + a*Cos[e + f*x])*EllipticE[(e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[(b + a*Cos[e + f
*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + ((a - b)*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(e + f*x)/2,
(2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) + (2*b*g*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]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*
Cos[e + f*x])*Sqrt[g*Sec[e + f*x]]*Sin[e + f*x])/(f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])

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Rubi [A]  time = 0.9223, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.282, Rules used = {3971, 3859, 2807, 2805, 3975, 2768, 2752, 2663, 2661, 2655, 2653} \[ -\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g (a-b) \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} F\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}}+\frac{2 b g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(b + a*Cos[e + f*x])*EllipticE[(e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[(b + a*Cos[e + f
*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + ((a - b)*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(e + f*x)/2,
(2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) + (2*b*g*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]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*
Cos[e + f*x])*Sqrt[g*Sec[e + f*x]]*Sin[e + f*x])/(f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])

Rule 3971

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

Rule 3859

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(d*Sqr
t[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 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]

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 2768

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b
^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x])), x] + Dist[d/(a*(b*c - a*
d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2752

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

Rule 2663

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

Rule 2661

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

Rule 2655

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

Rule 2653

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

Rubi steps

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

Mathematica [F]  time = 18.9168, size = 0, normalized size = 0. \[ \int \frac{(g \sec (e+f x))^{3/2} \sqrt{a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 0.355, size = 292, normalized size = 1. \begin{align*}{\frac{i \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{fc \left ( a\cos \left ( fx+e \right ) +b \right ) }\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ( 2\,a{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) -2\,b{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) -a{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) -b{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) +4\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-1,i\sqrt{{\frac{a-b}{a+b}}} \right ) b \right ) \sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}} \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}{\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))^(1/2)/(c+c*sec(f*x+e)),x)

[Out]

I/c/f*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(2*a*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b
))^(1/2))-2*b*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b))^(1/2))-a*EllipticE(I*(-1+cos(f*x+e))/sin(f
*x+e),(-(a-b)/(a+b))^(1/2))-b*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b))^(1/2))+4*EllipticPi(I*(-1+
cos(f*x+e))/sin(f*x+e),-1,I*((a-b)/(a+b))^(1/2))*b)*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*(g/cos(f*x+e))^(3/2)
*cos(f*x+e)^2/(a*cos(f*x+e)+b)/(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{\sqrt{b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{c \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))^(1/2)/(c+c*sec(f*x+e)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [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))**(1/2)/(c+c*sec(f*x+e)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{c \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))^(1/2)/(c+c*sec(f*x+e)),x, algorithm="giac")

[Out]

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