3.5.23 \(\int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) [423]

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

[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)

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Rubi [A]
time = 0.10, 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 = {2705, 3856, 2719} \begin {gather*} -\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)}} \end {gather*}

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 2705

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-a)*b*(a*Cs
c[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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3856

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*}

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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.70 \begin {gather*} -\frac {\left (-21+7 \csc ^2(e+f x)+2 \csc ^4(e+f x)+12 \csc ^6(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 )\right ) \tan (e+f x)}{60 f \sqrt {b \sec (e+f x)}} \end {gather*}

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]])

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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 918, normalized size = 7.46

method	result	size
default	\(\text {Expression too large to display}\)	\(918\)

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,method=_RETURNVERBOSE)

[Out]

-1/60/f*(-1+cos(f*x+e))^2*(-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)+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)-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*(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)^5+21
*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)-21*I*(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)
*sin(f*x+e)*cos(f*x+e)^4+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)^4-21*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)*(1/(cos
(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+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)-21*I*(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)*sin(f*x+e)+42*I*EllipticE(I*(-1+c
os(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*(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)*sin(f*
x+e)*cos(f*x+e)^5+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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 216, normalized size = 1.76 \begin {gather*} -\frac {21 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 56 \, \cos \left (f x + e\right )^{4} + 47 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{120 \, {\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )} \sin \left (f x + e\right )} \end {gather*}

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]

-1/120*(21*sqrt(2)*(I*cos(f*x + e)^4 - 2*I*cos(f*x + e)^2 + I)*sqrt(b)*sin(f*x + e)*weierstrassZeta(-4, 0, wei
erstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*sqrt(2)*(-I*cos(f*x + e)^4 + 2*I*cos(f*x + e)^2 -
I)*sqrt(b)*sin(f*x + e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2
*(21*cos(f*x + e)^6 - 56*cos(f*x + e)^4 + 47*cos(f*x + e)^2)*sqrt(b/cos(f*x + e)))/((b*f*cos(f*x + e)^4 - 2*b*
f*cos(f*x + e)^2 + b*f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{6}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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