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

Optimal. Leaf size=141 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} \sqrt{c} f}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} f \sqrt{c-d}} \]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c
]*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e +
f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)

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Rubi [A]  time = 0.338635, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {3938, 3934, 203, 3983} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} \sqrt{c} f}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} f \sqrt{c-d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c
]*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e +
f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)

Rule 3938

Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)]), x_Symbol] :> Di
st[1/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] - Dist[b/a, Int[Csc[e + f*x]/(Sqrt[a + b
*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3934

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3983

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx &=\frac{\int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx}{a}-\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+a c x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}\right )}{f}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{2+(a c-a d) x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}\right )}{f}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} \sqrt{c} f}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}\right )}{\sqrt{a} \sqrt{c-d} f}\\ \end{align*}

Mathematica [A]  time = 0.342739, size = 171, normalized size = 1.21 \[ \frac{2 \cos \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{c \cos (e+f x)+d} \left (\sqrt{2} \sqrt{c-d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sin \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c \cos (e+f x)+d}}\right )-\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c-d} \sin \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c \cos (e+f x)+d}}\right )\right )}{\sqrt{c} f \sqrt{c-d} \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[2]*Sqrt[c]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]] - Sqrt[c]*ArcTan[(
Sqrt[c - d]*Sin[(e + f*x)/2])/Sqrt[d + c*Cos[e + f*x]]])*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*Sec[e + f*x
])/(Sqrt[c]*Sqrt[c - d]*f*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])

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Maple [B]  time = 0.332, size = 424, normalized size = 3. \begin{align*} -2\,{\frac{\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) }{af\sqrt{c-d} \left ({c}^{2}-2\,cd+{d}^{2} \right ) c \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}} \left ( \sqrt{2}\sqrt{- \left ( c-d \right ) ^{4}c}\arctan \left ({\frac{c \left ( c-d \right ) ^{2}\sqrt{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sqrt{- \left ( c-d \right ) ^{4}c}\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}}} \right ) \sqrt{c-d}+\ln \left ( -{\frac{1}{\sin \left ( fx+e \right ) } \left ( \sqrt{c-d}\cos \left ( fx+e \right ) -\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sqrt{c-d} \right ) } \right ){c}^{3}-2\,\ln \left ( -{\frac{1}{\sin \left ( fx+e \right ) } \left ( \sqrt{c-d}\cos \left ( fx+e \right ) -\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sqrt{c-d} \right ) } \right ){c}^{2}d+\ln \left ( -{\frac{1}{\sin \left ( fx+e \right ) } \left ( \sqrt{c-d}\cos \left ( fx+e \right ) -\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sqrt{c-d} \right ) } \right ) c{d}^{2} \right ){\frac{1}{\sqrt{-2\,{\frac{d+c\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/f/a/(c-d)^(1/2)/(c^2-2*c*d+d^2)/c*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)
*cos(f*x+e)*(-1+cos(f*x+e))*(2^(1/2)*(-(c-d)^4*c)^(1/2)*arctan((c-d)^2*c*2^(1/2)/(-(c-d)^4*c)^(1/2)*(-1+cos(f*
x+e))/(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)/sin(f*x+e))*(c-d)^(1/2)+ln(-((c-d)^(1/2)*cos(f*x+e)-(-2*(d+c*
cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-(c-d)^(1/2))/sin(f*x+e))*c^3-2*ln(-((c-d)^(1/2)*cos(f*x+e)-(-2*(d
+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-(c-d)^(1/2))/sin(f*x+e))*c^2*d+ln(-((c-d)^(1/2)*cos(f*x+e)-(-2
*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-(c-d)^(1/2))/sin(f*x+e))*c*d^2)/sin(f*x+e)^2/(-2*(d+c*cos(f
*x+e))/(1+cos(f*x+e)))^(1/2)

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

[Out]

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

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Fricas [A]  time = 3.86766, size = 2290, normalized size = 16.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(2)*a*c*sqrt(-1/(a*c - a*d))*log((2*sqrt(2)*(c - d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*
cos(f*x + e) + d)/cos(f*x + e))*sqrt(-1/(a*c - a*d))*cos(f*x + e)*sin(f*x + e) + (3*c - d)*cos(f*x + e)^2 + 2*
(c + d)*cos(f*x + e) - c + 3*d)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) - 2*sqrt(-a*c)*log((2*a*c*cos(f*x + e)^
2 + 2*sqrt(-a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*
sin(f*x + e) - a*c + a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e) + 1)))/(a*c*f), 1/2*(sqrt(2)*a*c*sqrt(-1/(a
*c - a*d))*log((2*sqrt(2)*(c - d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x +
e))*sqrt(-1/(a*c - a*d))*cos(f*x + e)*sin(f*x + e) + (3*c - d)*cos(f*x + e)^2 + 2*(c + d)*cos(f*x + e) - c + 3
*d)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) - 4*sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x +
e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(a*c*sin(f*x + e))))/(a*c*f), (sqrt(2)*a*c*arctan(sqr
t(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(sqrt(a*c -
a*d)*sin(f*x + e)))/sqrt(a*c - a*d) - sqrt(-a*c)*log((2*a*c*cos(f*x + e)^2 + 2*sqrt(-a*c)*sqrt((a*cos(f*x + e)
+ a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - a*c + a*d + (a*c + a*d
)*cos(f*x + e))/(cos(f*x + e) + 1)))/(a*c*f), (sqrt(2)*a*c*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x +
e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(sqrt(a*c - a*d)*sin(f*x + e)))/sqrt(a*c - a*d) - 2*s
qrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(
f*x + e)/(a*c*sin(f*x + e))))/(a*c*f)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out