\(\int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx\) [2]

Optimal result

Integrand size = 37, antiderivative size = 209 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2987, 2986, 1232} \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} \sqrt {d} \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \cos (e+f x)}}{\sqrt {d} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{f \sqrt {b^2-a^2} \sqrt {g \sin (e+f x)}} \]

[In]

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

[Out]

(2*Sqrt[2]*Sqrt[d]*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]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[d]*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]])/(Sqrt[-a^2 + b^2]*f*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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sin (e+f x)} \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {\sin (e+f x)}} \, dx}{\sqrt {g \sin (e+f x)}} \\ & = -\frac {\left (2 \sqrt {2} \left (1-\frac {b}{\sqrt {-a^2+b^2}}\right ) d \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 )}{f \sqrt {g \sin (e+f x)}}-\frac {\left (2 \sqrt {2} \left (1+\frac {b}{\sqrt {-a^2+b^2}}\right ) d \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 )}{f \sqrt {g \sin (e+f x)}} \\ & = \frac {2 \sqrt {2} \sqrt {d} \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)}}{\sqrt {-a^2+b^2} f \sqrt {g \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {d} \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)}}{\sqrt {-a^2+b^2} f \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 = 17.92 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.78 \[ \int \frac {\sqrt {d \cos (e+f x)}}{(a+b \cos (e+f x)) \sqrt {g \sin (e+f x)}} \, dx=\frac {2 \sqrt {d \cos (e+f x)} \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {\tan (e+f x)} \left (\frac {\sqrt {a} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )\right )}{4 \sqrt {2} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \sqrt {\tan (e+f x)}}{\sqrt {\sec ^2(e+f x)} \left (a^2-b^2+a^2 \tan ^2(e+f x)\right ) \left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+2 \left (2 a^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right )\right ) \tan ^2(e+f x)\right )}\right )}{f (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {g \sin (e+f x)}} \]

[In]

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

[Out]

(2*Sqrt[d*Cos[e + f*x]]*(b + a*Sqrt[Sec[e + f*x]^2])*Sqrt[Tan[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt
[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/
4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]] + Log[Sqrt[
a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(
3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))]*Sqrt
[Tan[e + f*x]])/(Sqrt[Sec[e + f*x]^2]*(a^2 - b^2 + a^2*Tan[e + f*x]^2)*(-5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5
/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, -Tan[e + f*x]^
2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, -Tan[e + f*x]^2, -((a^2*Tan[e
+ f*x]^2)/(a^2 - b^2))])*Tan[e + f*x]^2))))/(f*(a + b*Cos[e + f*x])*Sqrt[Sec[e + f*x]^2]*Sqrt[g*Sin[e + f*x]]
)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(173)=346\).

Time = 4.77 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.43

method	result	size
default	\(\frac {\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \left (2 \sqrt {-a^{2}+b^{2}}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\cos \left (f x +e \right ) d}\, \left (\sec \left (f x +e \right )+1\right ) \sqrt {2}\, a}{f \sqrt {g \sin \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (b +\sqrt {-a^{2}+b^{2}}-a \right )}\)	\(508\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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