Integrand size = 21, antiderivative size = 102 \[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\csc (e+f x)}{6 b f \sqrt {b \sec (e+f x)}}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}-\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 b^2 f} \]
1/6*csc(f*x+e)/b/f/(b*sec(f*x+e))^(1/2)-1/3*csc(f*x+e)^3/b/f/(b*sec(f*x+e) )^(1/2)-1/6*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin( 1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(b*sec(f*x+e))^(1/2)/b^2/f
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.61 \[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\csc (e+f x)-2 \csc ^3(e+f x)-\frac {\operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{\sqrt {\cos (e+f x)}}}{6 b f \sqrt {b \sec (e+f x)}} \]
(Csc[e + f*x] - 2*Csc[e + f*x]^3 - EllipticF[(e + f*x)/2, 2]/Sqrt[Cos[e + f*x]])/(6*b*f*Sqrt[b*Sec[e + f*x]])
Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3103, 3042, 3105, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc (e+f x)^4}{(b \sec (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 3103 |
\(\displaystyle -\frac {\int \csc ^2(e+f x) \sqrt {b \sec (e+f x)}dx}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \csc (e+f x)^2 \sqrt {b \sec (e+f x)}dx}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle -\frac {\frac {1}{2} \int \sqrt {b \sec (e+f x)}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{2} \int \sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle -\frac {\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f}-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}}{6 b^2}-\frac {\csc ^3(e+f x)}{3 b f \sqrt {b \sec (e+f x)}}\) |
-1/3*Csc[e + f*x]^3/(b*f*Sqrt[b*Sec[e + f*x]]) - (-((b*Csc[e + f*x])/(f*Sq rt[b*Sec[e + f*x]])) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[ b*Sec[e + f*x]])/f)/(6*b^2)
3.5.35.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-a)*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n + 1)/(f*b*(m - 1))), x] + Simp[a^2*((n + 1)/(b^2*(m - 1))) Int[(a*Csc[e + f *x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && IntegersQ[2*m, 2*n]
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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {-i \left (\sin ^{2}\left (f x +e \right )\right ) F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sin \left (f x +e \right ) \tan \left (f x +e \right )+\cos \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )}{6 f \sqrt {b \sec \left (f x +e \right )}\, b \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(181\) |
1/6/f/(b*sec(f*x+e))^(1/2)/b/(cos(f*x+e)^2-1)*(-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)-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)*tan(f*x+e)+cos(f*x+e)* cot(f*x+e)+csc(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (b^{2} f \cos \left (f x + e\right )^{2} - b^{2} f\right )} \sin \left (f x + e\right )} \]
1/12*(sqrt(2)*(I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*weierstrassPInve rse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + sqrt(2)*(-I*cos(f*x + e)^2 + I )*sqrt(b)*sin(f*x + e)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) + 2*(cos(f*x + e)^3 + cos(f*x + e))*sqrt(b/cos(f*x + e)))/((b^2*f*c os(f*x + e)^2 - b^2*f)*sin(f*x + e))
\[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {\csc ^{4}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^4\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]