Integrand size = 21, antiderivative size = 95 \[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \csc (e+f x)}{2 f (b \sec (e+f x))^{3/2}}-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]
-1/2*b*csc(f*x+e)/f/(b*sec(f*x+e))^(3/2)-1/3*b*csc(f*x+e)^3/f/(b*sec(f*x+e ))^(3/2)-1/2*(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)
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\left (-3+\csc ^2(e+f x)+2 \csc ^4(e+f x)+3 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \tan (e+f x)}{6 f \sqrt {b \sec (e+f x)}} \]
-1/6*((-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])/(f*Sqrt[b*Sec[e + f*x]])
Time = 0.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {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.
\(\displaystyle \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc (e+f x)^4}{\sqrt {b \sec (e+f x)}}dx\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {1}{2} \int \frac {\csc ^2(e+f x)}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {\csc (e+f x)^2}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\sqrt {b \sec (e+f x)}}dx-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \sqrt {\cos (e+f x)}dx}{2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}\) |
-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
3.5.22.3.1 Defintions of rubi rules used
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 = 0.46 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.21
method | result | size |
default | \(\frac {3 i \left (\sin ^{2}\left (f x +e \right )\right ) E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 i \left (\sin ^{2}\left (f x +e \right )\right ) F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sin \left (f x +e \right ) \tan \left (f x +e \right )-3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sin \left (f x +e \right ) \tan \left (f x +e \right )+3 \sin \left (f x +e \right )+2 \cot \left (f x +e \right )}{6 f \sqrt {b \sec \left (f x +e \right )}\, \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(305\) |
1/6/f/(b*sec(f*x+e))^(1/2)/(cos(f*x+e)^2-1)*(3*I*sin(f*x+e)^2*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)-3*I*sin(f*x+e)^2*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)+3*I*(1/(cos(f*x+e)+ 1))^(1/2)*(cos(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)*tan(f*x+e)-3*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)*ta n(f*x+e)+3*sin(f*x+e)+2*cot(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.66 \[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {3 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (f x + e\right )^{4} - 5 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (b f \cos \left (f x + e\right )^{2} - b f\right )} \sin \left (f x + e\right )} \]
-1/12*(3*sqrt(2)*(I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*weierstrassZe ta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*s qrt(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*(3*cos(f*x + e)^4 - 5*cos(f*x + e)^2)*sqrt(b/cos(f*x + e)))/((b*f*cos(f*x + e)^2 - b *f)*sin(f*x + e))
\[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\csc ^{4}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^4\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]