3.274 \(\int \frac{\sec ^2(a+b x)}{\sqrt{\csc (a+b x)}} \, dx\)

Optimal. Leaf size=62 \[ \frac{\sec (a+b x)}{b \csc ^{\frac{3}{2}}(a+b x)}-\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]

[Out]

Sec[a + b*x]/(b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])
/b

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Rubi [A]  time = 0.0477514, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2626, 3771, 2639} \[ \frac{\sec (a+b x)}{b \csc ^{\frac{3}{2}}(a+b x)}-\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^2/Sqrt[Csc[a + b*x]],x]

[Out]

Sec[a + b*x]/(b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])
/b

Rule 2626

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

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

Rubi steps

\begin{align*} \int \frac{\sec ^2(a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=\frac{\sec (a+b x)}{b \csc ^{\frac{3}{2}}(a+b x)}-\frac{1}{2} \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{\sec (a+b x)}{b \csc ^{\frac{3}{2}}(a+b x)}-\frac{1}{2} \left (\sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx\\ &=\frac{\sec (a+b x)}{b \csc ^{\frac{3}{2}}(a+b x)}-\frac{\sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{b}\\ \end{align*}

Mathematica [A]  time = 0.14791, size = 54, normalized size = 0.87 \[ \frac{\sqrt{\csc (a+b x)} \left (\sin (a+b x) \tan (a+b x)+\sqrt{\sin (a+b x)} E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^2/Sqrt[Csc[a + b*x]],x]

[Out]

(Sqrt[Csc[a + b*x]]*(EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Sin[a + b*x]*Tan[a + b*x]))/b

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Maple [B]  time = 1.586, size = 177, normalized size = 2.9 \begin{align*}{\frac{1}{2\,\cos \left ( bx+a \right ) b}\sqrt{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) } \left ( 2\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{-\sin \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) -1 \right ) \left ( \sin \left ( bx+a \right ) +1 \right ) }}}{\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(b*x+a)^2*sin(b*x+a))^(1/2)*(2*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Ellipt
icE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Ellipti
cF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-2*cos(b*x+a)^2+2)/(-sin(b*x+a)*(sin(b*x+a)-1)*(sin(b*x+a)+1))^(1/2)/cos(b
*x+a)/sin(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right )^{2}}{\sqrt{\csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^2/sqrt(csc(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (b x + a\right )^{2}}{\sqrt{\csc \left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sec(b*x + a)^2/sqrt(csc(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (a + b x \right )}}{\sqrt{\csc{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(a + b*x)**2/sqrt(csc(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right )^{2}}{\sqrt{\csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^2/sqrt(csc(b*x + a)), x)