3.1.3 \(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx\) [3]

Optimal. Leaf size=208 \[ -\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2985, 2984, 504, 1232} \begin {gather*} \frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt {b-a} \sqrt {a+b} \sqrt {d \cos (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sq
rt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[g]*Sq
rt[Cos[e + f*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*
x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]])

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 2984

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

Rule 2985

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x
_)])), x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[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*} \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx &=\frac {\sqrt {\cos (e+f x)} \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \cos (e+f x))} \, dx}{\sqrt {d \cos (e+f x)}}\\ &=\frac {\left (4 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{f \sqrt {d \cos (e+f x)}}\\ &=\frac {\left (2 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{\sqrt {-a+b} f \sqrt {d \cos (e+f x)}}-\frac {\left (2 \sqrt {2} g \sqrt {\cos (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \sin (e+f x)}}{\sqrt {1+\cos (e+f x)}}\right )}{\sqrt {-a+b} f \sqrt {d \cos (e+f x)}}\\ &=-\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 12.99, size = 375, normalized size = 1.80 \begin {gather*} \frac {2 \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {g \sin (e+f x)} \left (\frac {-2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \text {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 )}{4 \sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(e+f x)}{3 \left (-a^2+b^2\right )}\right )}{f \sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(b + a*Sqrt[Sec[e + f*x]^2])*Sqrt[g*Sin[e + f*x]]*((-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]*Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3
/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))]*Tan[e + f*x]^(3/2))/(3*(-a^2 + b^2))))/
(f*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])*Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(164)=328\).
time = 0.25, size = 545, normalized size = 2.62

method	result	size
default	\(\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )-1}{\sin \left (f x +e \right )}}\, \left (\sqrt {-a^{2}+b^{2}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -a \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \right ) \sqrt {g \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) \sqrt {2}}{f \left (\cos \left (f x +e \right )-1\right ) \sqrt {d \cos \left (f x +e \right )}\, \left (\sqrt {-a^{2}+b^{2}}-a +b \right ) \left (\sqrt {-a^{2}+b^{2}}+a -b \right )}\)	\(545\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {d \cos {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {d\,\cos \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________