3.448 \(\int \frac{\csc ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}} \]

[Out]

Csc[e + f*x]/(2*b*f*(b*Sec[e + f*x])^(3/2)) - Csc[e + f*x]^3/(3*b*f*(b*Sec[e + f*x])^(3/2)) + EllipticE[(e + f
*x)/2, 2]/(2*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.108161, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2623, 2625, 3771, 2639} \[ \frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]

[Out]

Csc[e + f*x]/(2*b*f*(b*Sec[e + f*x])^(3/2)) - Csc[e + f*x]^3/(3*b*f*(b*Sec[e + f*x])^(3/2)) + EllipticE[(e + f
*x)/2, 2]/(2*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])

Rule 2623

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

Rule 2625

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*(m - 1)), x] + Dist[(a^2*(m + n - 2))/(m - 1), Int[(a*Csc[e +
f*x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] &&
!GtQ[n, m]

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{\csc ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}-\frac{\int \frac{\csc ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx}{2 b^2}\\ &=\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{\int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx}{4 b^2}\\ &=\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{\int \sqrt{\cos (e+f x)} \, dx}{4 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.25234, size = 79, normalized size = 0.77 \[ \frac{\sin (e+f x) \sqrt{b \sec (e+f x)} \left (-2 \csc ^4(e+f x)+5 \csc ^2(e+f x)+3 \sqrt{\cos (e+f x)} \csc (e+f x) E\left (\left .\frac{1}{2} (e+f x)\right |2\right )-3\right )}{6 b^3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]

[Out]

((-3 + 5*Csc[e + f*x]^2 - 2*Csc[e + f*x]^4 + 3*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Sqrt
[b*Sec[e + f*x]]*Sin[e + f*x])/(6*b^3*f)

________________________________________________________________________________________

Maple [C]  time = 0.162, size = 623, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^4/(b*sec(f*x+e))^(5/2),x)

[Out]

-1/6/f*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*(3*I*cos(f*x+e)^3*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(c
os(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-3*I*cos(f*x+e)^3*sin(f*x+e)*(1/(cos(f*x+e)+1))^(
1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*I*EllipticF(I*(-1+cos(f*x+e
))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2*sin(f*x+e)-3*I*Ellipt
icE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2*si
n(f*x+e)-3*I*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*
x+e)+1))^(1/2)*sin(f*x+e)+3*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)*sin(f*x+e)*(1/(cos(f*x+e)+1
))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-3*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1
/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+3*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*(1/(
cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+3*cos(f*x+e)^3+2*cos(f*x+e)^2-3*cos(f*x+e))/cos(f*x+e)^
3/sin(f*x+e)^7/(b/cos(f*x+e))^(5/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^4/(b*sec(f*x + e))^(5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4}}{b^{3} \sec \left (f x + e\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(f*x + e))*csc(f*x + e)^4/(b^3*sec(f*x + e)^3), 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(f*x+e)**4/(b*sec(f*x+e))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(csc(f*x + e)^4/(b*sec(f*x + e))^(5/2), x)