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

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]

[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] (verified)

Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2705, 3856, 2719} \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]

[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*} \text {integral}& = -\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] (verified)

Time = 0.38 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\left (-21+7 \csc ^2(e+f x)+2 \csc ^4(e+f x)+12 \csc ^6(e+f x)+21 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \tan (e+f x)}{60 f \sqrt {b \sec (e+f x)}} \]

[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]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.71

method	result	size
default	\(-\frac {21 i \left (\sin ^{4}\left (f x +e \right )\right ) E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-21 i \left (\sin ^{4}\left (f x +e \right )\right ) F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )-21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )+21 \left (\sin ^{3}\left (f x +e \right )\right )+14 \sin \left (f x +e \right ) \cos \left (f x +e \right )+12 \cot \left (f x +e \right )}{60 f \left (\cos \left (f x +e \right )-1\right )^{2} \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {b \sec \left (f x +e \right )}}\)	\(333\)

[In]

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

[Out]

-1/60/f/(cos(f*x+e)-1)^2/(cos(f*x+e)+1)^2/(b*sec(f*x+e))^(1/2)*(21*I*sin(f*x+e)^4*EllipticE(I*(-cot(f*x+e)+csc
(f*x+e)),I)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(cos(f*x+e)+1))^(1/2)-21*I*sin(f*x+e)^4*EllipticF(I*(-cot(f*x
+e)+csc(f*x+e)),I)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(cos(f*x+e)+1))^(1/2)+21*I*(1/(cos(f*x+e)+1))^(1/2)*(c
os(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-cot(f*x+e)+csc(f*x+e)),I)*sin(f*x+e)^3*tan(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*(-cot(f*x+e)+csc(f*x+e)),I)*sin(f*x+e)^3*tan(f*x
+e)+21*sin(f*x+e)^3+14*sin(f*x+e)*cos(f*x+e)+12*cot(f*x+e))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {21 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 56 \, \cos \left (f x + e\right )^{4} + 47 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{120 \, {\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )} \sin \left (f x + e\right )} \]

[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))

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^6\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

[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)