3.146 \(\int \frac{1}{(a+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx\)

Optimal. Leaf size=319 \[ \frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right ),\frac{c+d}{c-d}\right )}{a f (c-d)}-\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \Pi \left (\frac{c+d}{c};\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right )}{a c f}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{c-d}{c+d}\right )}{a f (c-d) \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]

[Out]

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

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Rubi [A]  time = 0.369961, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3929, 3921, 3784, 3832, 3968} \[ \frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right )}{a f (c-d)}-\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} \Pi \left (\frac{c+d}{c};\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right )}{a c f}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{c-d}{c+d}\right )}{a f (c-d) \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

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

Rule 3929

Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Dist[1
/(c*(b*c - a*d)), Int[(b*c - a*d - b*d*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d^2/(c*(b*c - a*d
)), Int[(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 3921

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[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 3784

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^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{1}{(a+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx &=-\frac{\int \frac{-a c+a d-a d \sec (e+f x)}{\sqrt{c+d \sec (e+f x)}} \, dx}{a^2 (c-d)}+\frac{a \int \frac{\sec (e+f x) \sqrt{c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{-a c+a d}\\ &=-\frac{E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac{c-d}{c+d}\right ) \sqrt{\frac{1}{1+\sec (e+f x)}} \sqrt{c+d \sec (e+f x)}}{a (c-d) f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}+\frac{\int \frac{1}{\sqrt{c+d \sec (e+f x)}} \, dx}{a}+\frac{d \int \frac{\sec (e+f x)}{\sqrt{c+d \sec (e+f x)}} \, dx}{a (c-d)}\\ &=\frac{2 \sqrt{c+d} \cot (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right ) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac{2 \sqrt{c+d} \cot (e+f x) \Pi \left (\frac{c+d}{c};\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right ) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac{E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac{c-d}{c+d}\right ) \sqrt{\frac{1}{1+\sec (e+f x)}} \sqrt{c+d \sec (e+f x)}}{a (c-d) f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}\\ \end{align*}

Mathematica [A]  time = 5.37417, size = 193, normalized size = 0.61 \[ \frac{4 \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left (2 (c-2 d) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{c-d}{c+d}\right )+(c+d) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )+4 (c-d) \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )\right )}{a f (d-c) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} (\cos (e+f x)+1)^2 \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

(4*Cos[(e + f*x)/2]^4*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*((c + d)*EllipticE[ArcSin[Tan[(e
+ f*x)/2]], (c - d)/(c + d)] + 2*(c - 2*d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 4*(c - d)*E
llipticPi[-1, -ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]))/(a*(-c + d)*f*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])
]*(1 + Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]])

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Maple [A]  time = 0.351, size = 327, normalized size = 1. \begin{align*} -{\frac{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{fa \left ( c-d \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \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{c-d}{c+d}}} \right ) c-4\,{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) d+c{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) +d{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) -4\,c{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{c-d}{c+d}}} \right ) +4\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{c-d}{c+d}}} \right ) d \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/f/(c-d)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(
1+cos(f*x+e)))^(1/2)*(1+cos(f*x+e))^2*(-1+cos(f*x+e))*(2*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1
/2))*c-4*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1/2))*d+c*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((
c-d)/(c+d))^(1/2))+d*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((c-d)/(c+d))^(1/2))-4*c*EllipticPi((-1+cos(f*x+e))/
sin(f*x+e),-1,((c-d)/(c+d))^(1/2))+4*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((c-d)/(c+d))^(1/2))*d)/(d+c*cos
(f*x+e))/sin(f*x+e)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \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(1/(a+a*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((a*sec(f*x + e) + a)*sqrt(d*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{d \sec \left (f x + e\right ) + c}}{a d \sec \left (f x + e\right )^{2} + a c +{\left (a c + a d\right )} \sec \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{c + d \sec{\left (e + f x \right )}} \sec{\left (e + f x \right )} + \sqrt{c + d \sec{\left (e + f x \right )}}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(c + d*sec(e + f*x))*sec(e + f*x) + sqrt(c + d*sec(e + f*x))), x)/a

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

[Out]

integrate(1/((a*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)), x)