3.275 \(\int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx\)

Optimal. Leaf size=209 \[ \frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{c f (a-b)} \]

[Out]

(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*
(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*c*f) + (EllipticE[ArcSin[Tan[e
+ f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[a + b*Sec[e + f*x]])/((a - b)*
c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])

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Rubi [A]  time = 0.28782, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3972, 3832, 3968} \[ \frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{c f (a-b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*
(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*c*f) + (EllipticE[ArcSin[Tan[e
+ f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[a + b*Sec[e + f*x]])/((a - b)*
c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])

Rule 3972

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
), x_Symbol] :> Dist[b/(b*c - a*d), Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[d/(b*c - a*d), In
t[(Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]])/(c + d*Csc[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] || EqQ[c^2 - d^2, 0])

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3968

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
), x_Symbol] :> -Simp[(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c/(c + d*Csc[e + f*x])]*EllipticE[ArcSin[(c*Cot[e + f*x])
/(c + d*Csc[e + f*x])], -((b*c - a*d)/(b*c + a*d))])/(d*f*Sqrt[(c*d*(a + b*Csc[e + f*x]))/((b*c + a*d)*(c + d*
Csc[e + f*x]))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^
2, 0]

Rubi steps

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

Mathematica [B]  time = 17.7217, size = 2173, normalized size = 10.4 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*Sec[e + f*x]^2*((2*Sin[e + f*x])/(-a + b) - (2*Tan[(e + f*x)/2])/(-
a + b)))/(f*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) - (2*Cos[e/2 + (f*x)/2]^2*(-(b/((-a + b)*Sqrt[b + a
*Cos[e + f*x]]*Sqrt[Sec[e + f*x]])) - (a*Sqrt[Sec[e + f*x]])/((-a + b)*Sqrt[b + a*Cos[e + f*x]]) + (b*Sqrt[Sec
[e + f*x]])/((-a + b)*Sqrt[b + a*Cos[e + f*x]]) - (a*Cos[2*(e + f*x)]*Sqrt[Sec[e + f*x]])/((-a + b)*Sqrt[b + a
*Cos[e + f*x]]))*Sec[e + f*x]^(3/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((a - b)*EllipticE[ArcSin[Sqrt[(a -
b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt
[2]*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + T
an[(e + f*x)/2]^2))/(((a - b)/(a + b))^(3/2)*(a + b)*f*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[a + b*Sec[e
+ f*x]]*(c + c*Sec[e + f*x])*((-2*Sec[(e + f*x)/2]^2*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Tan[(e + f*x)/2]*((
a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*S
ec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e
+ f*x])*Tan[(e + f*x)/2]))/(((a - b)/(a + b))^(3/2)*(a + b)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[Cos[e + f*x]*Sec[(e
+ f*x)/2]^4]) - (a*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sin[e + f*x]*((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/
(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]
*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[
(e + f*x)/2]^2))/(((a - b)/(a + b))^(3/2)*(a + b)*(b + a*Cos[e + f*x])^(3/2)*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2
]^4]) + (Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2
]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]*Sqrt[(a - b)/(a + b)]*S
qrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-(Sec[(e + f*x)/2]^4*Sin[e + f*x]
) + 2*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2))/(((a - b)/(a + b))^(3/2)*(a
+ b)*Sqrt[b + a*Cos[e + f*x]]*(Cos[e + f*x]*Sec[(e + f*x)/2]^4)^(3/2)) - (2*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f
*x]]*(-1 + Tan[(e + f*x)/2]^2)*((Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b + a*Cos[e + f*
x])*Sec[(e + f*x)/2]^2)/Sqrt[2] - Sqrt[2]*a*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sin[e
+ f*x]*Tan[(e + f*x)/2] + (Sqrt[(a - b)/(a + b)]*(b + a*Cos[e + f*x])*((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[e
+ f*x])^2 - Sin[e + f*x]/(1 + Cos[e + f*x]))*Tan[(e + f*x)/2])/(Sqrt[2]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])])
+ ((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*(-((a*Sec[(e + f*x)/2]^
2*Sin[e + f*x])/(a + b)) + ((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(a + b)))/(2*Sqrt[((b +
a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)]) + ((a - b)*Sqrt[(a - b)/(a + b)]*Sec[(e + f*x)/2]^2*Sqrt[((b + a
*Cos[e + f*x])*Sec[(e + f*x)/2]^2)/(a + b)]*Sqrt[1 - Tan[(e + f*x)/2]^2])/(2*Sqrt[1 - ((a - b)*Tan[(e + f*x)/2
]^2)/(a + b)])))/(((a - b)/(a + b))^(3/2)*(a + b)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^
4]) - (((a - b)*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[e
+ f*x])*Sec[(e + f*x)/2]^2)/(a + b)] + Sqrt[2]*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(b
+ a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2)*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2
]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(((a - b)/(a + b))^(3/2)*(a + b)*Sqrt[b + a*Cos[e + f*x]]*
Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^4]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))

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Maple [A]  time = 0.293, size = 225, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{fc \left ( a-b \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ( 2\,{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) b-a{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) -b{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c/f/(a-b)*(1+cos(f*x+e))^2*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)
*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(-1+cos(f*x+e))*(2*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^
(1/2))*b-a*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))-b*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((
a-b)/(a+b))^(1/2)))/(a*cos(f*x+e)+b)/sin(f*x+e)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{b \sec \left (f x + e\right ) + a}{\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(f*x + e)/(sqrt(b*sec(f*x + e) + a)*(c*sec(f*x + e) + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{b c \sec \left (f x + e\right )^{2} +{\left (a + b\right )} c \sec \left (f x + e\right ) + a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec{\left (e + f x \right )}}{\sqrt{a + b \sec{\left (e + f x \right )}} \sec{\left (e + f x \right )} + \sqrt{a + b \sec{\left (e + f x \right )}}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(e + f*x)/(sqrt(a + b*sec(e + f*x))*sec(e + f*x) + sqrt(a + b*sec(e + f*x))), x)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{b \sec \left (f x + e\right ) + a}{\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(f*x + e)/(sqrt(b*sec(f*x + e) + a)*(c*sec(f*x + e) + c)), x)