Integrand size = 21, antiderivative size = 126 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {8 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac {8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}} \]
-4/39*b*sin(f*x+e)/f/(b*sec(f*x+e))^(7/2)+8/195*sin(f*x+e)/b/f/(b*sec(f*x+ e))^(3/2)-2/13*b*sin(f*x+e)^3/f/(b*sec(f*x+e))^(7/2)+8/65*(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))/b^2/ f/cos(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2)
Time = 0.57 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {192 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\cos ^{\frac {3}{2}}(e+f x) (-6 \sin (e+f x)-55 \sin (3 (e+f x))+15 \sin (5 (e+f x)))}{1560 f \cos ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{5/2}} \]
(192*EllipticE[(e + f*x)/2, 2] + Cos[e + f*x]^(3/2)*(-6*Sin[e + f*x] - 55* Sin[3*(e + f*x)] + 15*Sin[5*(e + f*x)]))/(1560*f*Cos[e + f*x]^(5/2)*(b*Sec [e + f*x])^(5/2))
Time = 0.68 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3107, 3042, 3107, 3042, 4256, 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.
| \(\displaystyle \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc (e+f x)^4 (b \sec (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {6}{13} \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}}dx-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{13} \int \frac {1}{\csc (e+f x)^2 (b \sec (e+f x))^{5/2}}dx-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \int \frac {1}{(b \sec (e+f x))^{5/2}}dx-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \int \frac {1}{\left (b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \left (\frac {3 \int \frac {1}{\sqrt {b \sec (e+f x)}}dx}{5 b^2}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \left (\frac {3 \int \sqrt {\cos (e+f x)}dx}{5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \left (\frac {3 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {6}{13} \left (\frac {2}{9} \left (\frac {6 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\right )-\frac {2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\) |
(-2*b*Sin[e + f*x]^3)/(13*f*(b*Sec[e + f*x])^(7/2)) + (6*((-2*b*Sin[e + f* x])/(9*f*(b*Sec[e + f*x])^(7/2)) + (2*((6*EllipticE[(e + f*x)/2, 2])/(5*b^ 2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*Sin[e + f*x])/(5*b*f*(b* Sec[e + f*x])^(3/2))))/9))/13
3.5.44.3.1 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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
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 = 2.64 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.86
| method | result | size |
| default | \(\frac {\frac {2 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{13}+\frac {2 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{13}+\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 )}{65}-\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 )}{65}-\frac {10 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{39}+\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 )}{65}-\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 )}{65}-\frac {10 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{39}+\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 )}{65}-\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 )}{65}+\frac {8 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{195}+\frac {8 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{195}+\frac {8 \sin \left (f x +e \right )}{65}}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {b \sec \left (f x +e \right )}\, b^{2}}\) | \(486\) |
2/195/f/(cos(f*x+e)+1)/(b*sec(f*x+e))^(1/2)/b^2*(15*cos(f*x+e)^6*sin(f*x+e )+15*cos(f*x+e)^5*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)*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)-25*cos(f*x+e)^4*sin(f*x+e)+24*I*(1/(c os(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)-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)-25*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)+4*sin(f*x+e)*cos(f*x+e)^2+4*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 = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 \, {\left ({\left (15 \, \cos \left (f x + e\right )^{6} - 25 \, \cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 6 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )\right )}}{195 \, b^{3} f} \]
2/195*((15*cos(f*x + e)^6 - 25*cos(f*x + e)^4 + 4*cos(f*x + e)^2)*sqrt(b/c os(f*x + e))*sin(f*x + e) + 6*I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, wei erstrassPInverse(-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^3*f)
\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {\sin ^{4}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]