Integrand size = 21, antiderivative size = 95 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
-5/6*b*csc(f*x+e)/f/(b*sec(f*x+e))^(1/2)-1/3*b*csc(f*x+e)^3/f/(b*sec(f*x+e ))^(1/2)+5/6*(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)/f
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (-\cot (e+f x) \left (5+2 \csc ^2(e+f x)\right )+5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
((-(Cot[e + f*x]*(5 + 2*Csc[e + f*x]^2)) + 5*Sqrt[Cos[e + f*x]]*EllipticF[ (e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]])/(6*f)
Time = 0.47 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, 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, 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.
\(\displaystyle \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^4 \sqrt {b \sec (e+f x)}dx\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {5}{6} \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \int \csc (e+f x)^2 \sqrt {b \sec (e+f x)}dx-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \int \sqrt {b \sec (e+f x)}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \int \sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {5}{6} \left (\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f}-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}\) |
-1/3*(b*Csc[e + f*x]^3)/(f*Sqrt[b*Sec[e + f*x]]) + (5*(-((b*Csc[e + f*x])/ (f*Sqrt[b*Sec[e + f*x]])) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]* Sqrt[b*Sec[e + f*x]])/f))/6
3.4.83.3.1 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[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 = 0.58 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {i \sqrt {b \sec \left (f x +e \right )}\, \left (-5 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-5 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{2}\left (f x +e \right )\right )+5 i \left (\cos ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )-7 i \cot \left (f x +e \right )\right )}{6 f \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(186\) |
1/6*I/f*(b*sec(f*x+e))^(1/2)/(cos(f*x+e)^2-1)*(-5*(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)^2*cos(f*x+e)-5*(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)^2+5*I*cos(f*x +e)^2*cot(f*x+e)-7*I*cot(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.56 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (5 \, \cos \left (f x + e\right )^{3} - 7 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
-1/12*(5*sqrt(2)*(I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*weierstrassPI nverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + 5*sqrt(2)*(-I*cos(f*x + e)^ 2 + I)*sqrt(b)*sin(f*x + e)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*si n(f*x + e)) + 2*(5*cos(f*x + e)^3 - 7*cos(f*x + e))*sqrt(b/cos(f*x + e)))/ ((f*cos(f*x + e)^2 - f)*sin(f*x + e))
\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \]
\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \]
\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^4} \,d x \]