3.1432 \(\int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx\)

Optimal. Leaf size=209 \[ \frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.354131, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {2908, 2907, 1218} \[ \frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2908

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 2907

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 1218

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx &=\frac{\sqrt{\cos (e+f x)} \int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{\cos (e+f x)} (a+b \sin (e+f x))} \, dx}{\sqrt{g \cos (e+f x)}}\\ &=\frac{\left (2 \sqrt{2} \left (1-\frac{b}{\sqrt{-a^2+b^2}}\right ) d \sqrt{\cos (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (b-\sqrt{-a^2+b^2}\right ) d+a x^2\right ) \sqrt{1-\frac{x^4}{d^2}}} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{1+\cos (e+f x)}}\right )}{f \sqrt{g \cos (e+f x)}}+\frac{\left (2 \sqrt{2} \left (1+\frac{b}{\sqrt{-a^2+b^2}}\right ) d \sqrt{\cos (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (b+\sqrt{-a^2+b^2}\right ) d+a x^2\right ) \sqrt{1-\frac{x^4}{d^2}}} \, dx,x,\frac{\sqrt{d \sin (e+f x)}}{\sqrt{1+\cos (e+f x)}}\right )}{f \sqrt{g \cos (e+f x)}}\\ &=-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt{-a^2+b^2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{-a^2+b^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{1+\cos (e+f x)}}\right )\right |-1\right )}{\sqrt{-a^2+b^2} f \sqrt{g \cos (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 4.20825, size = 170, normalized size = 0.81 \[ \frac{2 \sqrt{2} \sqrt{\tan \left (\frac{1}{2} (e+f x)\right )} \cot (e+f x) \sqrt{d \sin (e+f x)} \left (\Pi \left (\frac{a}{\sqrt{b^2-a^2}-b};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (e+f x)\right )}\right )\right |-1\right )-\Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (e+f x)\right )}\right )\right |-1\right )\right )}{f \sqrt{b^2-a^2} \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{g \cos (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.224, size = 520, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}a\sin \left ( fx+e \right ) }{f \left ( -1+\cos \left ( fx+e \right ) \right ) }\sqrt{d\sin \left ( fx+e \right ) } \left ( \sqrt{-{a}^{2}+{b}^{2}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{a \left ( a-b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{-{a}^{2}+{b}^{2}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},-{a \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{a \left ( a-b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) a-{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{a \left ( a-b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) b-{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},-{a \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) a+{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},-{a \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) b \right ) \sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}} \left ( a-b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1} \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) ^{-1}{\frac{1}{\sqrt{g\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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