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]

-(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^(1/2)/b/(d*
cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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*}

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Mathematica [C]  time = 0.06, 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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{c d \cos \left (b x + a\right ) \sin \left (b x + a\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

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)

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maple [B]  time = 0.10, size = 151, normalized size = 2.85 \[ -\frac {\EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \left (\sin ^{2}\left (b x +a \right )\right ) \sqrt {2}}{b \sqrt {c \sin \left (b x +a \right )}\, \left (-1+\cos \left (b x +a \right )\right ) \sqrt {d \cos \left (b x +a \right )}} \]

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*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)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {d\,\cos \left (a+b\,x\right )}\,\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \sin {\left (a + b x \right )}} \sqrt {d \cos {\left (a + b x \right )}}}\, dx \]

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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