3.278 \(\int \frac{\sec ^4(a+b x)}{\sqrt{\csc (a+b x)}} \, dx\)

Optimal. Leaf size=92 \[ \frac{\sec ^3(a+b x)}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{\sec (a+b x)}{2 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0803564, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2626, 3771, 2639} \[ \frac{\sec ^3(a+b x)}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{\sec (a+b x)}{2 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^4/Sqrt[Csc[a + b*x]],x]

[Out]

Sec[a + b*x]/(2*b*Csc[a + b*x]^(3/2)) + Sec[a + b*x]^3/(3*b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*Elliptic
E[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(2*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 ^4(a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=\frac{\sec ^3(a+b x)}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{1}{2} \int \frac{\sec ^2(a+b x)}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{\sec (a+b x)}{2 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{\sec ^3(a+b x)}{3 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{1}{4} \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{\sec (a+b x)}{2 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{\sec ^3(a+b x)}{3 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{1}{4} \left (\sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx\\ &=\frac{\sec (a+b x)}{2 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{\sec ^3(a+b x)}{3 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)}}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.241992, size = 76, normalized size = 0.83 \[ \frac{\cos (a+b x) \sqrt{\csc (a+b x)} \left (2 \sec ^4(a+b x)+\sec ^2(a+b x)+3 \sqrt{\sin (a+b x)} \sec (a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )-3\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^4/Sqrt[Csc[a + b*x]],x]

[Out]

(Cos[a + b*x]*Sqrt[Csc[a + b*x]]*(-3 + Sec[a + b*x]^2 + 2*Sec[a + b*x]^4 + 3*EllipticE[(-2*a + Pi - 2*b*x)/4,
2]*Sec[a + b*x]*Sqrt[Sin[a + b*x]]))/(6*b)

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Maple [A]  time = 2.687, size = 160, normalized size = 1.7 \begin{align*}{\frac{1}{12\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}b} \left ( 6\,\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 ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}-3\,\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},1/2\,\sqrt{2} \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}-6\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+4 \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)^4/csc(b*x+a)^(1/2),x)

[Out]

1/12/sin(b*x+a)^(1/2)/cos(b*x+a)^3*(6*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Ellipti
cE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)^2-3*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))
^(1/2)*EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)^2-6*cos(b*x+a)^4+2*cos(b*x+a)^2+4)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right )^{4}}{\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)^4/csc(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(b*x + a)^4/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 )^{4}}{\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)^4/csc(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sec(b*x + a)^4/sqrt(csc(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{4}{\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)**4/csc(b*x+a)**(1/2),x)

[Out]

Integral(sec(a + b*x)**4/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 )^{4}}{\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)^4/csc(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sec(b*x + a)^4/sqrt(csc(b*x + a)), x)