3.234 \(\int \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sqrt{\sin (2 a+2 b x)} \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0927454, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2630, 2573, 2641} \[ \frac{\sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]],x]

[Out]

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

Rule 2630

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Csc[e + f*
x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n, Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n),
x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]

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

Mathematica [C]  time = 0.670832, size = 68, normalized size = 1.28 \[ \frac{\tan ^3(a+b x) \left (-\cot ^2(a+b x)\right )^{7/4} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{3}{2},\csc ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]],x]

[Out]

((-Cot[a + b*x]^2)^(7/4)*Sqrt[d*Csc[a + b*x]]*Hypergeometric2F1[1/2, 3/4, 3/2, Csc[a + b*x]^2]*Sqrt[c*Sec[a +
b*x]]*Tan[a + b*x]^3)/b

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Maple [B]  time = 0.196, size = 157, normalized size = 3. \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{b \left ( -1+\cos \left ( bx+a \right ) \right ) }\sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}\sqrt{{\frac{d}{\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 ) }{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(1/2)*(c*sec(b*x+a))^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))**(1/2)*(c*sec(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(c*sec(a + b*x))*sqrt(d*csc(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a)), x)