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

Optimal. Leaf size=92 \[ \frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{6 b} \]

[Out]

(5*Sec[a + b*x])/(6*b*Sqrt[Csc[a + b*x]]) + Sec[a + b*x]^3/(3*b*Sqrt[Csc[a + b*x]]) + (5*Sqrt[Csc[a + b*x]]*El
lipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(6*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0813144, 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, 2641} \[ \frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sec[a + b*x])/(6*b*Sqrt[Csc[a + b*x]]) + Sec[a + b*x]^3/(3*b*Sqrt[Csc[a + b*x]]) + (5*Sqrt[Csc[a + b*x]]*El
lipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(6*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 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{\csc (a+b x)} \sec ^4(a+b x) \, dx &=\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5}{6} \int \sqrt{\csc (a+b x)} \sec ^2(a+b x) \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5}{12} \int \sqrt{\csc (a+b x)} \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{1}{12} \left (5 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{5 \sec (a+b x)}{6 b \sqrt{\csc (a+b x)}}+\frac{\sec ^3(a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{5 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.378396, size = 64, normalized size = 0.7 \[ \frac{\sqrt{\csc (a+b x)} \left (\tan (a+b x) \left (2 \sec ^2(a+b x)+5\right )-5 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Csc[a + b*x]]*(-5*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + (5 + 2*Sec[a + b*x]^2)*Tan[a
+ b*x]))/(6*b)

________________________________________________________________________________________

Maple [A]  time = 1.954, size = 168, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( 12\,\sin \left ( bx+a \right ) -12 \right ) \left ( \sin \left ( bx+a \right ) +1 \right ) \cos \left ( bx+a \right ) b}\sqrt{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) } \left ( 5\,\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}+10\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) +4\,\sin \left ( bx+a \right ) \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(csc(b*x+a)^(1/2)*sec(b*x+a)^4,x)

[Out]

-1/12*(cos(b*x+a)^2*sin(b*x+a))^(1/2)*(5*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Elli
pticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)^2+10*cos(b*x+a)^2*sin(b*x+a)+4*sin(b*x+a))/(sin(b*x+a)-1)/(
sin(b*x+a)+1)/(-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

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(1/2)*sec(b*x+a)^4,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**(1/2)*sec(b*x+a)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(1/2)*sec(b*x+a)^4,x, algorithm="giac")

[Out]

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