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

Optimal. Leaf size=213 \[ \frac{2 (a c-b d) \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{a f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right )}{a f} \]

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

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Rubi [A]  time = 0.384645, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2829, 3969, 3832, 3973} \[ \frac{2 (a c-b d) \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{a f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right )|\frac{c+d}{c-d}\right )}{a f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*Sec[e + f*x]]/(a + b*Cos[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*f) + (2*(a*c - b*d)*EllipticPi[(2*a)/(
a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/(c + d)]*Tan[e + f*x]
)/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])

Rule 2829

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int
[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !In
tegerQ[n] && IntegerQ[m]

Rule 3969

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
), x_Symbol] :> Dist[b/d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/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] && NeQ[a^2 - b^2, 0] && NeQ[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 3973

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

Rubi steps

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

Mathematica [A]  time = 3.83941, size = 186, normalized size = 0.87 \[ \frac{4 \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \sqrt{c+d \sec (e+f x)} \left (2 (b d-a c) \Pi \left (\frac{b-a}{a+b};-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )-(a+b) (c-d) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )\right )}{f (a-b) (a+b) (c \cos (e+f x)+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.33, size = 357, normalized size = 1.7 \begin{align*} -2\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( \cos \left ( fx+e \right ) -1 \right ) }{f \left ( a+b \right ) \left ( a-b \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 ) }{\cos \left ( fx+e \right ) +1}}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) }}} \left ({\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) ac-{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) ad+{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) bc-{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) bd-2\,{\it EllipticPi} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},\sqrt{{\frac{c-d}{c+d}}} \right ) ac+2\,{\it EllipticPi} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},\sqrt{{\frac{c-d}{c+d}}} \right ) bd \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/f/(a+b)/(a-b)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e
))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)+1)^2*(EllipticF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2))*a*c-Ellipt
icF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2))*a*d+EllipticF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2)
)*b*c-EllipticF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2))*b*d-2*EllipticPi((cos(f*x+e)-1)/sin(f*x+e),-(a-
b)/(a+b),((c-d)/(c+d))^(1/2))*a*c+2*EllipticPi((cos(f*x+e)-1)/sin(f*x+e),-(a-b)/(a+b),((c-d)/(c+d))^(1/2))*b*d
)*(cos(f*x+e)-1)/(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{\sqrt{d \sec \left (f x + e\right ) + c}}{b \cos \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*sec(f*x + e) + c)/(b*cos(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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