Integrand size = 21, antiderivative size = 67 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}} \] Output:
4/5*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec(f*x+e) )^(1/2)-2/5*b*sin(f*x+e)/f/(b*sec(f*x+e))^(3/2)
Time = 0.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {\sqrt {b \sec (e+f x)} \left (-8 \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sin (e+f x)+\sin (3 (e+f x))\right )}{10 b f} \] Input:
Integrate[Sin[e + f*x]^2/Sqrt[b*Sec[e + f*x]],x]
Output:
-1/10*(Sqrt[b*Sec[e + f*x]]*(-8*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + Sin[e + f*x] + Sin[3*(e + f*x)]))/(b*f)
Time = 0.57 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {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.
\(\displaystyle \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc (e+f x)^2 \sqrt {b \sec (e+f x)}}dx\) |
\(\Big \downarrow \) 3107 |
\(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 \int \sqrt {\cos (e+f x)}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\) |
Input:
Int[Sin[e + f*x]^2/Sqrt[b*Sec[e + f*x]],x]
Output:
(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))
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 = 3.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.87
method | result | size |
default | \(\frac {\frac {2 \sin \left (f x +e \right ) \left (-\cos \left (f x +e \right )^{2}-\cos \left (f x +e \right )+2\right )}{5}+\frac {4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (2+\cos \left (f x +e \right )+\sec \left (f x +e \right )\right ) \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{5}-\frac {4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (2+\cos \left (f x +e \right )+\sec \left (f x +e \right )\right ) \operatorname {EllipticE}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )}{5}}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {b \sec \left (f x +e \right )}}\) | \(192\) |
Input:
int(sin(f*x+e)^2/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/5/f/(cos(f*x+e)+1)/(b*sec(f*x+e))^(1/2)*(sin(f*x+e)*(-cos(f*x+e)^2-cos(f *x+e)+2)+2*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(2 +cos(f*x+e)+sec(f*x+e))*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)-2*I*(1/(cos (f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(2+cos(f*x+e)+sec(f*x+ e))*EllipticE(I*(cot(f*x+e)-csc(f*x+e)),I))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 \, {\left (\sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )\right )}}{5 \, b f} \] Input:
integrate(sin(f*x+e)^2/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/5*(sqrt(b/cos(f*x + e))*cos(f*x + e)^2*sin(f*x + e) - I*sqrt(2)*sqrt(b) *weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f* x + e))) + I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4 , 0, cos(f*x + e) - I*sin(f*x + e))))/(b*f)
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \] Input:
integrate(sin(f*x+e)**2/(b*sec(f*x+e))**(1/2),x)
Output:
Integral(sin(e + f*x)**2/sqrt(b*sec(e + f*x)), x)
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)^2/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sin(f*x + e)^2/sqrt(b*sec(f*x + e)), x)
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)^2/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sin(f*x + e)^2/sqrt(b*sec(f*x + e)), x)
Timed out. \[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(sin(e + f*x)^2/(b/cos(e + f*x))^(1/2),x)
Output:
int(sin(e + f*x)^2/(b/cos(e + f*x))^(1/2), x)
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \sin \left (f x +e \right )^{2}}{\sec \left (f x +e \right )}d x \right )}{b} \] Input:
int(sin(f*x+e)^2/(b*sec(f*x+e))^(1/2),x)
Output:
(sqrt(b)*int((sqrt(sec(e + f*x))*sin(e + f*x)**2)/sec(e + f*x),x))/b