Integrand size = 21, antiderivative size = 98 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}} \]
-2/9*b*sin(f*x+e)/f/(b*sec(f*x+e))^(7/2)+4/45*sin(f*x+e)/b/f/(b*sec(f*x+e) )^(3/2)+4/15*(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 = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {\frac {96 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}-4 \sin (2 (e+f x))-10 \sin (4 (e+f x))}{360 b^2 f \sqrt {b \sec (e+f x)}} \]
((96*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] - 4*Sin[2*(e + f*x)] - 10*Sin[4*(e + f*x)])/(360*b^2*f*Sqrt[b*Sec[e + f*x]])
Time = 0.47 (sec) , antiderivative size = 103, 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, 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.
| \(\displaystyle \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc (e+f x)^2 (b \sec (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{(b \sec (e+f x))^{5/2}}dx-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{\left (b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {2}{9} \left (\frac {3 \int \frac {1}{\sqrt {b \sec (e+f x)}}dx}{5 b^2}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {2}{9} \left (\frac {3 \int \sqrt {\cos (e+f x)}dx}{5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {3 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2}{9} \left (\frac {6 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}\) |
(-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
3.5.45.3.1 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 = 1.69 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.63
| method | result | size |
| default | \(-\frac {2 \left (6 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )-6 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )+5 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+12 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )-12 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )+5 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+6 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-6 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-2 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right ) \cos \left (f x +e \right )-6 \sin \left (f x +e \right )\right )}{45 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {b \sec \left (f x +e \right )}\, b^{2}}\) | \(454\) |
-2/45/f/(cos(f*x+e)+1)/(b*sec(f*x+e))^(1/2)/b^2*(6*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)*cos(f*x+e)-6*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)*cos(f*x+e)+5*cos(f*x+e)^4*sin( f*x+e)+12*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell ipticF(I*(-cot(f*x+e)+csc(f*x+e)),I)-12*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)+5*cos(f *x+e)^3*sin(f*x+e)+6*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)*sec(f*x+e)-6*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)+c sc(f*x+e)),I)*sec(f*x+e)-2*sin(f*x+e)*cos(f*x+e)^2-2*sin(f*x+e)*cos(f*x+e) -6*sin(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left ({\left (5 \, \cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 3 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )\right )}}{45 \, b^{3} f} \]
-2/45*((5*cos(f*x + e)^4 - 2*cos(f*x + e)^2)*sqrt(b/cos(f*x + e))*sin(f*x + e) - 3*I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*I*sqrt(2)*sqrt(b)*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(b^3*f)
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]