3.423 \(\int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\)

Optimal. Leaf size=123 \[ -\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]

[Out]

-7/20*b*csc(f*x+e)/f/(b*sec(f*x+e))^(3/2)-7/30*b*csc(f*x+e)^3/f/(b*sec(f*x+e))^(3/2)-1/5*b*csc(f*x+e)^5/f/(b*s
ec(f*x+e))^(3/2)-7/20*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/
cos(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2625, 3771, 2639} \[ -\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]

[Out]

(-7*b*Csc[e + f*x])/(20*f*(b*Sec[e + f*x])^(3/2)) - (7*b*Csc[e + f*x]^3)/(30*f*(b*Sec[e + f*x])^(3/2)) - (b*Cs
c[e + f*x]^5)/(5*f*(b*Sec[e + f*x])^(3/2)) - (7*EllipticE[(e + f*x)/2, 2])/(20*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 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]

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]

Rubi steps

\begin {align*} \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx &=-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7}{10} \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7}{20} \int \frac {\csc ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7}{40} \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 \int \sqrt {\cos (e+f x)} \, dx}{40 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 86, normalized size = 0.70 \[ -\frac {\tan (e+f x) \left (12 \csc ^6(e+f x)+2 \csc ^4(e+f x)+7 \csc ^2(e+f x)+21 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )-21\right )}{60 f \sqrt {b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]

[Out]

-1/60*((-21 + 7*Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 12*Csc[e + f*x]^6 + 21*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*Ell
ipticE[(e + f*x)/2, 2])*Tan[e + f*x])/(f*Sqrt[b*Sec[e + f*x]])

________________________________________________________________________________________

fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}}{b \sec \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(f*x + e))*csc(f*x + e)^6/(b*sec(f*x + e)), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(csc(f*x + e)^6/sqrt(b*sec(f*x + e)), x)

________________________________________________________________________________________

maple [C]  time = 0.25, size = 918, normalized size = 7.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x)

[Out]

-1/60/f*(-1+cos(f*x+e))^2*(21*I*(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)*sin(f*x+e)*cos(f*x+e)+42*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)-21*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)+21*I*EllipticF(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)-21*I*cos(f*x+e)^5*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)-21*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)^4*si
n(f*x+e)-21*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)+42*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)^3*sin(f*x+e)+21*I*cos(f*x+e)^5*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)-42*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)^3*sin(f*x+e)+21*I*EllipticF(I*(-1+co
s(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)^4*sin(f*x+e)-42*
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)+21*cos(f*x+e)^5-14*cos(f*x+e)^4-42*cos(f*x+e)^3+26*cos(f*x+e)^2+21*cos(f*x+e))*(cos(f*x+e)+1)
^2*(b/cos(f*x+e))^(1/2)/b/sin(f*x+e)^9

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^6/sqrt(b*sec(f*x + e)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^6\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2)),x)

[Out]

int(1/(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2)), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**6/(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________