3.422 \(\int \frac{\csc ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{b \csc (e+f x)}{2 f (b \sec (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.101027, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2625, 3771, 2639} \[ -\frac{b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{b \csc (e+f x)}{2 f (b \sec (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.208904, size = 74, normalized size = 0.78 \[ -\frac{\tan (e+f x) \left (2 \csc ^4(e+f x)+\csc ^2(e+f x)+3 \sqrt{\cos (e+f x)} \csc (e+f x) E\left (\left .\frac{1}{2} (e+f x)\right |2\right )-3\right )}{6 f \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-3 + 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])*Tan[e
+ f*x])/(6*f*Sqrt[b*Sec[e + f*x]])

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Maple [C]  time = 0.177, size = 618, normalized size = 6.5 \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))^(1/2),x)

[Out]

1/6/f*(-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)/(cos(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*EllipticE(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*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)/(c
os(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)+1)^2*(b/cos(f*x+e
))^(1/2)/b/sin(f*x+e)^7

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

[Out]

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

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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 \sec \left (f x + e\right )}, 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))^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(e + f*x)**4/sqrt(b*sec(e + f*x)), x)

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

[Out]

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