3.293 \(\int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]

[Out]

(EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])

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Rubi [A]  time = 0.0554808, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2573, 2641} \[ \frac{\sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])

Rule 2573

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 2641

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx &=\frac{\sqrt{\sin (2 a+2 b x)} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ &=\frac{F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)}}{b \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ \end{align*}

Mathematica [C]  time = 0.0573399, size = 65, normalized size = 1.23 \[ \frac{2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\sin ^2(a+b x)\right )}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*Sqrt[d*Cos[a + b*x
]]*Sqrt[c*Sin[a + b*x]])

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Maple [B]  time = 0.089, size = 151, normalized size = 2.9 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{b \left ( -1+\cos \left ( bx+a \right ) \right ) }\sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{d\cos \left ( bx+a \right ) }}}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*2^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((-1+cos(b*x+a))/sin(b*x+a))^
(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*sin(b*x+a)^2/
(c*sin(b*x+a))^(1/2)/(-1+cos(b*x+a))/(d*cos(b*x+a))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}{c d \cos \left (b x + a\right ) \sin \left (b x + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))/(c*d*cos(b*x + a)*sin(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \sin{\left (a + b x \right )}} \sqrt{d \cos{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*cos(b*x+a))**(1/2)/(c*sin(b*x+a))**(1/2),x)

[Out]

Integral(1/(sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))), x)