Integrand size = 21, antiderivative size = 95 \[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {8 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}} \]
-4/15*b*sin(f*x+e)/f/(b*sec(f*x+e))^(3/2)-2/9*b*sin(f*x+e)^3/f/(b*sec(f*x+ e))^(3/2)+8/15*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(s in(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 = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\frac {192 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}-68 \sin (2 (e+f x))+10 \sin (4 (e+f x))}{360 f \sqrt {b \sec (e+f x)}} \]
((192*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] - 68*Sin[2*(e + f*x)] + 10*Sin[4*(e + f*x)])/(360*f*Sqrt[b*Sec[e + f*x]])
Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3107, 3042, 3107, 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.
| \(\displaystyle \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc (e+f x)^4 \sqrt {b \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {2}{3} \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \int \frac {1}{\csc (e+f x)^2 \sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 \int \sqrt {\cos (e+f x)}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2}{3} \left (\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\) |
(-2*b*Sin[e + f*x]^3)/(9*f*(b*Sec[e + f*x])^(3/2)) + (2*((4*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x])/(5*f*(b*Sec[e + f*x])^(3/2))))/3
3.5.18.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Simp[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1) /(a*f*(m + n))), x] + Simp[(m + 1)/(a^2*(m + n)) Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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 = 1.53 (sec) , antiderivative size = 451, normalized size of antiderivative = 4.75
| method | result | size |
| default | \(\frac {\frac {8 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 ) \cos \left (f x +e \right )}{15}-\frac {8 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 ) \cos \left (f x +e \right )}{15}+\frac {2 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{9}+\frac {16 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 )}{15}-\frac {16 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 )}{15}+\frac {2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{9}+\frac {8 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 ) \sec \left (f x +e \right )}{15}-\frac {8 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 ) \sec \left (f x +e \right )}{15}-\frac {22 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{45}-\frac {22 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{45}+\frac {8 \sin \left (f x +e \right )}{15}}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {b \sec \left (f x +e \right )}}\) | \(451\) |
2/45/f/(cos(f*x+e)+1)/(b*sec(f*x+e))^(1/2)*(12*I*EllipticE(I*(-cot(f*x+e)+ csc(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)* cos(f*x+e)-12*I*EllipticF(I*(-cot(f*x+e)+csc(f*x+e)),I)*(1/(cos(f*x+e)+1)) ^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)+5*cos(f*x+e)^4*sin(f*x +e)+24*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ellipt icE(I*(-cot(f*x+e)+csc(f*x+e)),I)-24*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)+5*cos(f*x+ e)^3*sin(f*x+e)+12*I*EllipticE(I*(-cot(f*x+e)+csc(f*x+e)),I)*(1/(cos(f*x+e )+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sec(f*x+e)-12*I*EllipticF(I* (-cot(f*x+e)+csc(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+ e)+1))^(1/2)*sec(f*x+e)-11*sin(f*x+e)*cos(f*x+e)^2-11*sin(f*x+e)*cos(f*x+e )+12*sin(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 \, {\left ({\left (5 \, \cos \left (f x + e\right )^{4} - 11 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 6 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )\right )}}{45 \, b f} \]
2/45*((5*cos(f*x + e)^4 - 11*cos(f*x + e)^2)*sqrt(b/cos(f*x + e))*sin(f*x + e) + 6*I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 6*I*sqrt(2)*sqrt(b)*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(b*f)
\[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sin ^{4}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]