Integrand size = 21, antiderivative size = 123 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac {7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \] Output:
-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*sec(f*x+e))^(3/2)-7/20*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)
Time = 0.52 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\left (-21+7 \csc ^2(e+f x)+2 \csc ^4(e+f x)+12 \csc ^6(e+f x)+21 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \tan (e+f x)}{60 f \sqrt {b \sec (e+f x)}} \] Input:
Integrate[Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]
Output:
-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]*EllipticE[(e + f*x)/2, 2])*Tan[e + f*x])/ (f*Sqrt[b*Sec[e + f*x]])
Time = 1.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3105, 3042, 3105, 3042, 3105, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc (e+f x)^6}{\sqrt {b \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3105 |
| \(\displaystyle \frac {7}{10} \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \int \frac {\csc (e+f x)^4}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3105 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \int \frac {\csc ^2(e+f x)}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \int \frac {\csc (e+f x)^2}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3105 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \left (-\frac {\int \sqrt {\cos (e+f x)}dx}{2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \left (-\frac {\int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \left (-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
Input:
Int[Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]
Output:
-1/5*(b*Csc[e + f*x]^5)/(f*(b*Sec[e + f*x])^(3/2)) + (7*(-1/3*(b*Csc[e + f *x]^3)/(f*(b*Sec[e + f*x])^(3/2)) + (-((b*Csc[e + f*x])/(f*(b*Sec[e + f*x] )^(3/2))) - EllipticE[(e + f*x)/2, 2]/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]))/2))/10
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] + Simp[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]
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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Result contains complex when optimal does not.
Time = 3.58 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.51
| method | result | size |
| default | \(\frac {-\frac {i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (-21-21 \sec \left (f x +e \right )\right )}{60}-\frac {i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (21+21 \sec \left (f x +e \right )\right )}{60}-\frac {7 \csc \left (f x +e \right )}{20}-\frac {7 \cot \left (f x +e \right ) \csc \left (f x +e \right )^{2}}{30}-\frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )^{4}}{5}}{f \sqrt {b \sec \left (f x +e \right )}}\) | \(186\) |
Input:
int(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/f*(-1/60*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*El lipticE(I*(cot(f*x+e)-csc(f*x+e)),I)*(-21-21*sec(f*x+e))-1/60*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)-c sc(f*x+e)),I)*(21+21*sec(f*x+e))-7/20*csc(f*x+e)-7/30*cot(f*x+e)*csc(f*x+e )^2-1/5*cot(f*x+e)*csc(f*x+e)^4)/(b*sec(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {21 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 56 \, \cos \left (f x + e\right )^{4} + 47 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{120 \, {\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )} \sin \left (f x + e\right )} \] Input:
integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-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, weierstrassPInverse(-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, c os(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*co s(f*x + e)^2 + b*f)*sin(f*x + e))
\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\csc ^{6}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \] Input:
integrate(csc(f*x+e)**6/(b*sec(f*x+e))**(1/2),x)
Output:
Integral(csc(e + f*x)**6/sqrt(b*sec(e + f*x)), x)
\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(csc(f*x + e)^6/sqrt(b*sec(f*x + e)), x)
\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(csc(f*x + e)^6/sqrt(b*sec(f*x + e)), x)
Timed out. \[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^6\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(1/(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2)),x)
Output:
int(1/(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2)), x)
\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \csc \left (f x +e \right )^{6}}{\sec \left (f x +e \right )}d x \right )}{b} \] Input:
int(csc(f*x+e)^6/(b*sec(f*x+e))^(1/2),x)
Output:
(sqrt(b)*int((sqrt(sec(e + f*x))*csc(e + f*x)**6)/sec(e + f*x),x))/b