3.184 \(\int \sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx\)
Optimal. Leaf size=123 \[ \frac{2 \sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{f}+\frac{2 \sqrt{a} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{f} \]
[Out]
(2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])
/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*
x]])])/f
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Rubi [A] time = 0.338634, antiderivative size = 123, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used =
{3932, 3934, 203, 3980, 206} \[ \frac{2 \sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{f}+\frac{2 \sqrt{a} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{c+d \sec (e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]
[Out]
(2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])
/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*
x]])])/f
Rule 3932
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Dist[c
, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] + Dist[d, 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 3980
Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Dist[(-2*b)/f, Subst[Int[1/(1 - b*d*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sq
rt[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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \, dx &=c \int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx+d \int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx\\ &=-\frac{(2 a c) \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 a d) \operatorname{Subst}\left (\int \frac{1}{1-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 \sqrt{a} \sqrt{c} \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 )}{f}+\frac{2 \sqrt{a} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c+d \sec (e+f x)}}\right )}{f}\\ \end{align*}
Mathematica [A] time = 18.1594, size = 240, normalized size = 1.95 \[ -\frac{2 \cot (e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c+d \sec (e+f x)} \left (\sqrt{c^2 \sin ^2(e+f x)} \sqrt{c-c \cos (e+f x)} \tan ^{-1}\left (\frac{\sqrt{c (\cos (e+f x)+1)} \sqrt{c \cos (e+f x)+d}}{\sqrt{c^2 \sin ^2(e+f x)}}\right )-2 \sqrt{c} \sqrt{d} \sin ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{c (\cos (e+f x)+1)} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{c \cos (e+f x)+d}}{\sqrt{d} \sqrt{c-c \cos (e+f x)}}\right )\right )}{f \sqrt{c (\cos (e+f x)+1)} \sqrt{c-c \cos (e+f x)} \sqrt{c \cos (e+f x)+d}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]
[Out]
(-2*Cot[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]]*(-2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt
[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])]*Sqrt[c*(1 + Cos[e + f*x])]*Sin[(e + f*x)/2]^2 + ArcT
an[(Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[d + c*Cos[e + f*x]])/Sqrt[c^2*Sin[e + f*x]^2]]*Sqrt[c - c*Cos[e + f*x]]*Sq
rt[c^2*Sin[e + f*x]^2]))/(f*Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])
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Maple [B] time = 0.42, size = 1563, normalized size = 12.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*sec(f*x+e))^(1/2)*(a+a*sec(f*x+e))^(1/2),x)
[Out]
-1/f*2^(1/2)/(c-d)^(1/2)/(c^2-2*c*d+d^2)/(-d)^(1/2)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(1/cos(f*x+e)*a*(1+cos
(f*x+e)))^(1/2)*cos(f*x+e)*(-1+cos(f*x+e))*(2^(1/2)*(-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^3-3*2^(1/2)*(-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^2*d+3*2^(1/2)*(-d)^(1/2)*l
n(-((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^
2-2^(1/2)*(-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))*d^3-2^(1/2)*(-d)^(1/2)*ln(1/(c-d)^(1/2)*((-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(c-d)^(1
/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/sin(f*x+e))*c^3+3*2^(1/2)*(-d)^(1/2)*ln(1/(c-d)^(1/2)*((-2*(d+c*
cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(c-d)^(1/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/sin(f*x+e))*c^2*d-3*2^
(1/2)*(-d)^(1/2)*ln(1/(c-d)^(1/2)*((-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(c-d)^(1/2)*sin(f*x+e)-c*cos(f*x
+e)+d*cos(f*x+e)+c-d)/sin(f*x+e))*c*d^2+2^(1/2)*(-d)^(1/2)*ln(1/(c-d)^(1/2)*((-2*(d+c*cos(f*x+e))/(1+cos(f*x+e
)))^(1/2)*(c-d)^(1/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/sin(f*x+e))*d^3+(c-d)^(1/2)*ln(2*(2^(1/2)*(-d)
^(1/2)*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+
e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*c^2*d-2*(c-d)^(1/2)*ln(2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(1+cos(f*
x+e)))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*c*
d^2+(c-d)^(1/2)*ln(2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*
sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*d^3-(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)
*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d
)/(-1+cos(f*x+e)+sin(f*x+e)))*c^2*d+2*(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)
))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/(-1+cos(f*x+e)+sin(f*x+e)))*c*d^2
-(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-c*sin(f*x+e)-d*si
n(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/(-1+cos(f*x+e)+sin(f*x+e)))*d^3+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)*
(-d)^(1/2))/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 \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((c+d*sec(f*x+e))^(1/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(a*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c), x)
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Fricas [A] time = 1.40323, size = 2034, normalized size = 16.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sec(f*x+e))^(1/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[(sqrt(a*d)*log((2*sqrt(a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*c
os(f*x + e)*sin(f*x + e) + (a*c - a*d)*cos(f*x + e)^2 + 2*a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e)^2 + co
s(f*x + e))) + 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))*sqr
t((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)))/f, -(2*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))) - sqrt(a*d)*log((2*sqrt(a*d)*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)*cos(f*x + e)^2 + 2*a
*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e)^2 + cos(f*x + e))))/f, -(2*sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt((a*c
os(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(a*d*sin(f*x + e))) - 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)))/f, -2
*(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))*c
os(f*x + e)/(a*c*sin(f*x + e))) + sqrt(-a*d)*arctan(sqrt(-a*d)*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*d*sin(f*x + e))))/f]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((c+d*sec(f*x+e))**(1/2)*(a+a*sec(f*x+e))**(1/2),x)
[Out]
Integral(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((c+d*sec(f*x+e))^(1/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")
[Out]
Timed out