3.3.7 \(\int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx\) [207]

3.3.7.1 Optimal result

Integrand size = 29, antiderivative size = 389 \[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} f}+\frac {2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {\frac {a+b}{c+d}} \sqrt {c+d \sec (e+f x)}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {\frac {a+b}{c+d}} f} \]

output

2*cot(f*x+e)*EllipticPi(((a+b)/(c+d))^(1/2)*(c+d*sec(f*x+e))^(1/2)/(a+b*se 
c(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec( 
f*x+e))*(-(-a*d+b*c)*(1-sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^(1/2)*((-a*d+b 
*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)/f/((a+b)/(c+d))^(1/2)-2*c 
ot(f*x+e)*EllipticPi((a+b)^(1/2)*(c+d*sec(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*s 
ec(f*x+e))^(1/2),a*(c+d)/(a+b)/c,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec 
(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sec(f*x+e))/(c+d)/(a+b*sec(f*x+e)))^( 
1/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)/f/(a+b)^(1/2 
)
 

3.3.7.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.77 (sec) , antiderivative size = 40517, normalized size of antiderivative = 104.16 \[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\text {Result too large to show} \]

input

Integrate[Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]
 

output

Result too large to show
 

3.3.7.3 Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 4420, 3042, 4424, 4470}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 4420

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

\(\Big \downarrow \) 3042

\(\displaystyle c \int \frac {\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4424

\(\displaystyle d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f \sqrt {a+b}}\)

\(\Big \downarrow \) 4470

\(\displaystyle \frac {2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {\frac {a+b}{c+d}} \sqrt {c+d \sec (e+f x)}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f \sqrt {\frac {a+b}{c+d}}}-\frac {2 \sqrt {c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f \sqrt {a+b}}\)

input

Int[Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]
 

output

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

3.3.7.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4420

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

rule 4424

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] 
*(d_.) + (c_)], x_Symbol] :> Simp[2*((a + b*Csc[e + f*x])/(c*f*Rt[(a + b)/( 
c + d), 2]*Cot[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Csc[e + f*x])/((c - d)*(a 
+ b*Csc[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Csc[e + f*x])/((c + d)*(a + 
b*Csc[e + f*x])))]*EllipticPi[a*((c + d)/(c*(a + b))), ArcSin[Rt[(a + b)/(c 
 + d), 2]*(Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]])], (a - b)*((c 
 + d)/((a + b)*(c - d)))], 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 4470

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

3.3.7.4 Maple [A] (verified)

Time = 11.87 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {2 \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \left (2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -\frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) a c -\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) a c +\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) a d +\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) b c -\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) b d +2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) b d \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{f \sqrt {\frac {a -b}{a +b}}\, \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(510\)

input

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

output

-2/f/((a-b)/(a+b))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(c+d*sec(f*x+e))^(1/2)*(1/ 
(c+d)*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*EllipticPi(((a-b)/(a+b))^( 
1/2)*(cot(f*x+e)-csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b) 
)^(1/2))*a*c-EllipticF(((a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(f*x+e)),((a+b)* 
(c-d)/(a-b)/(c+d))^(1/2))*a*c+EllipticF(((a-b)/(a+b))^(1/2)*(cot(f*x+e)-cs 
c(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*d+EllipticF(((a-b)/(a+b))^(1/ 
2)*(cot(f*x+e)-csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b*c-EllipticF( 
((a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2 
))*b*d+2*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(f*x+e)-csc(f*x+e)),(a+b)/(a-b 
),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*b*d)*(1/(a+b)*(b+a*cos(f*x+e))/ 
(cos(f*x+e)+1))^(1/2)/(d+c*cos(f*x+e))/(b+a*cos(f*x+e))*(cos(f*x+e)^2+cos( 
f*x+e))
 

3.3.7.5 Fricas [F(-1)]

Timed out. \[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.3.7.6 Sympy [F]

\[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int \sqrt {a + b \sec {\left (e + f x \right )}} \sqrt {c + d \sec {\left (e + f x \right )}}\, dx \]

input

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

output

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

3.3.7.7 Maxima [F]

\[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c} \,d x } \]

input

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

output

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

3.3.7.8 Giac [F]

\[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c} \,d x } \]

input

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

output

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

3.3.7.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int \sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]

input

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

output

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