\(\int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx\) [272]

Optimal result

Integrand size = 39, antiderivative size = 168 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a f \sqrt {c+d \sec (e+f x)}}+\frac {2 (a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a (a+b) f \sqrt {c+d \sec (e+f x)}} \]

[Out]

2*d*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2,2^(1/2)*(c/(c+d))^(1/2))*(
(d+c*cos(f*x+e))/(c+d))^(1/2)*(g*sec(f*x+e))^(1/2)/a/f/(c+d*sec(f*x+e))^(1/2)+2*(a*c-b*d)*(cos(1/2*f*x+1/2*e)^
2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(a+b),2^(1/2)*(c/(c+d))^(1/2))*((d+c*cos(f*x+e))
/(c+d))^(1/2)*(g*sec(f*x+e))^(1/2)/a/(a+b)/f/(c+d*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3041, 4056, 3944, 2886, 2884, 4060} \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f (a+b) \sqrt {c+d \sec (e+f x)}}+\frac {2 d \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}} \]

[In]

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

[Out]

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

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3041

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

Rule 3944

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[d*Sqrt
[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f
*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4056

Int[((csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)
]*(d_.) + (c_)), x_Symbol] :> Dist[b/d, Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*
c - a*d)/d, Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x], x] /; FreeQ[{a, b,
c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 4060

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))), x_Symbol] :> Dist[g*Sqrt[g*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]),
Int[1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(g \sec (e+f x))^{3/2} \sqrt {c+d \sec (e+f x)}}{b+a \sec (e+f x)} \, dx}{g} \\ & = \frac {d \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a g}+\frac {(a c-b d) \int \frac {(g \sec (e+f x))^{3/2}}{(b+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx}{a g} \\ & = \frac {\left (d \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {d+c \cos (e+f x)}} \, dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {\left ((a c-b d) \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(a+b \cos (e+f x)) \sqrt {d+c \cos (e+f x)}} \, dx}{a \sqrt {c+d \sec (e+f x)}} \\ & = \frac {\left (d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {\left ((a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(a+b \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a f \sqrt {c+d \sec (e+f x)}}+\frac {2 (a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a (a+b) f \sqrt {c+d \sec (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=-\frac {2 i \sqrt {-\frac {c (-1+\cos (e+f x))}{c+d}} \sqrt {\frac {c (1+\cos (e+f x))}{c-d}} \cot (e+f x) \left (\operatorname {EllipticPi}\left (1-\frac {c}{d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )-\operatorname {EllipticPi}\left (\frac {b (-c+d)}{-a c+b d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )\right ) \sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a \sqrt {\frac {1}{c-d}} f \sqrt {d+c \cos (e+f x)}} \]

[In]

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

[Out]

((-2*I)*Sqrt[-((c*(-1 + Cos[e + f*x]))/(c + d))]*Sqrt[(c*(1 + Cos[e + f*x]))/(c - d)]*Cot[e + f*x]*(EllipticPi
[1 - c/d, I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)] - EllipticPi[(b*(-c + d))/
(-(a*c) + b*d), I*ArcSinh[Sqrt[(c - d)^(-1)]*Sqrt[d + c*Cos[e + f*x]]], (-c + d)/(c + d)])*Sqrt[g*Sec[e + f*x]
]*Sqrt[c + d*Sec[e + f*x]])/(a*Sqrt[(c - d)^(-1)]*f*Sqrt[d + c*Cos[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.63

method	result	size
default	\(\frac {2 i \sqrt {g \sec \left (f x +e \right )}\, \cos \left (f x +e \right ) \sqrt {c +d \sec \left (f x +e \right )}\, \left (\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} d +\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b d +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) a^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) a b c +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f a \left (a -b \right ) \left (a +b \right ) \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\)	\(442\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((g*sec(f*x+e))^(1/2)*(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)*sqrt(g*sec(f*x + e))/(b*cos(f*x + e) + a), x)

Giac [F]

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

[In]

integrate((g*sec(f*x+e))^(1/2)*(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)*sqrt(g*sec(f*x + e))/(b*cos(f*x + e) + a), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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