\(\int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx\) [4]

Optimal result

Integrand size = 37, antiderivative size = 273 \[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=-\frac {2 \sqrt {2} b \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a^2+b^2} \sqrt {d} f \sqrt {g \sin (e+f x)}}+\frac {2 \sqrt {2} b \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a^2+b^2} \sqrt {d} f \sqrt {g \sin (e+f x)}}+\frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}} \]

[Out]

-2*b*EllipticPi((d*cos(f*x+e))^(1/2)/d^(1/2)/(1+sin(f*x+e))^(1/2),-a/(b-(-a^2+b^2)^(1/2)),I)*2^(1/2)*sin(f*x+e
)^(1/2)/a/f/(-a^2+b^2)^(1/2)/d^(1/2)/(g*sin(f*x+e))^(1/2)+2*b*EllipticPi((d*cos(f*x+e))^(1/2)/d^(1/2)/(1+sin(f
*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)),I)*2^(1/2)*sin(f*x+e)^(1/2)/a/f/(-a^2+b^2)^(1/2)/d^(1/2)/(g*sin(f*x+e))^(
1/2)-(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*sin(2*f*x+2*e)^(1/2)/a
/f/(d*cos(f*x+e))^(1/2)/(g*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2989, 2653, 2720, 2987, 2986, 1232} \[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=-\frac {2 \sqrt {2} b \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{a \sqrt {d} f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}}+\frac {2 \sqrt {2} b \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{a \sqrt {d} f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}}+\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}} \]

[In]

Int[1/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]

[Out]

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

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2986

Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]))
, x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Dist[2*Sqrt[2]*d*((b + q)/(f*q)), Subst[Int[1/((d*(b + q) + a*x^2
)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Dist[2*Sqrt[2]*d*((b - q)/(f*q
)), Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]],
x]] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2987

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

Rule 2989

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[1/a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] - Dist[b/(a*d), Int[(g*Cos[
e + f*x])^p*((d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2
- b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && LtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}} \, dx}{a}-\frac {b \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx}{a d} \\ & = -\frac {\left (b \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {\sin (e+f x)}} \, dx}{a d \sqrt {g \sin (e+f x)}}+\frac {\sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{a \sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}} \\ & = \frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}+\frac {\left (2 \sqrt {2} b \left (1-\frac {b}{\sqrt {-a^2+b^2}}\right ) \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-\left (\left (-b+\sqrt {-a^2+b^2}\right ) d\right )+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a f \sqrt {g \sin (e+f x)}}+\frac {\left (2 \sqrt {2} b \left (1+\frac {b}{\sqrt {-a^2+b^2}}\right ) \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-\left (\left (-b-\sqrt {-a^2+b^2}\right ) d\right )+a x^2\right ) \sqrt {1-\frac {x^4}{d^2}}} \, dx,x,\frac {\sqrt {d \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a f \sqrt {g \sin (e+f x)}} \\ & = -\frac {2 \sqrt {2} b \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a^2+b^2} \sqrt {d} f \sqrt {g \sin (e+f x)}}+\frac {2 \sqrt {2} b \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a^2+b^2} \sqrt {d} f \sqrt {g \sin (e+f x)}}+\frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 16.82 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {18 (a+b) \sqrt {g \sin (e+f x)} \left (5 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+\operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f g \sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \left (45 (a+b) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+\tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (45 (a+b) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )-36 (a-b) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+18 (a+b) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+10 \left (-2 (a-b) \operatorname {AppellF1}\left (\frac {9}{4},\frac {1}{2},2,\frac {13}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+(a+b) \operatorname {AppellF1}\left (\frac {9}{4},\frac {3}{2},1,\frac {13}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(-a+b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]

[In]

Integrate[1/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[g*Sin[e + f*x]]),x]

[Out]

(18*(a + b)*Sqrt[g*Sin[e + f*x]]*(5*AppellF1[1/4, 1/2, 1, 5/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^
2)/(a + b)] + AppellF1[5/4, 1/2, 1, 9/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)]*Tan[(e + f
*x)/2]^2))/(f*g*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*(45*(a + b)*AppellF1[1/4, 1/2, 1, 5/4, Tan[(e + f*x)
/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + Tan[(e + f*x)/2]^2*(45*(a + b)*AppellF1[5/4, 1/2, 1, 9/4, Tan[
(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] - 36*(a - b)*AppellF1[5/4, 1/2, 2, 9/4, Tan[(e + f*x)/2
]^2, ((-a + b)*Tan[(e + f*x)/2]^2)/(a + b)] + 18*(a + b)*AppellF1[5/4, 3/2, 1, 9/4, Tan[(e + f*x)/2]^2, ((-a +
b)*Tan[(e + f*x)/2]^2)/(a + b)] + 10*(-2*(a - b)*AppellF1[9/4, 1/2, 2, 13/4, Tan[(e + f*x)/2]^2, ((-a + b)*Ta
n[(e + f*x)/2]^2)/(a + b)] + (a + b)*AppellF1[9/4, 3/2, 1, 13/4, Tan[(e + f*x)/2]^2, ((-a + b)*Tan[(e + f*x)/2
]^2)/(a + b)])*Tan[(e + f*x)/2]^2)))

Maple [A] (warning: unable to verify)

Time = 4.84 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.88

method	result	size
default	\(\frac {\left (\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}-2 F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-a^{2}+b^{2}}+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) a b -\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b^{2}-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) a b +\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b^{2}\right ) \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (1+\cos \left (f x +e \right )\right ) \sqrt {2}}{f \sqrt {g \sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right ) d}\, \left (b +\sqrt {-a^{2}+b^{2}}-a \right ) \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \sqrt {-a^{2}+b^{2}}}\)	\(514\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {d \cos {\left (e + f x \right )}} \sqrt {g \sin {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(d*cos(e + f*x))*sqrt(g*sin(e + f*x))*(a + b*cos(e + f*x))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/((b*cos(f*x + e) + a)*sqrt(d*cos(f*x + e))*sqrt(g*sin(f*x + e))), x)

Giac [F]

\[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/((b*cos(f*x + e) + a)*sqrt(d*cos(f*x + e))*sqrt(g*sin(f*x + e))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {d\,\cos \left (e+f\,x\right )}\,\sqrt {g\,\sin \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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