Integrand size = 21, antiderivative size = 72 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {6 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
-2/5*d*cos(f*x+e)/f/(d*csc(f*x+e))^(3/2)-6/5*(sin(1/2*e+1/4*Pi+1/2*f*x)^2) ^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/ 2))/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {-\frac {12 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}-2 \sin (2 (e+f x))}{10 f \sqrt {d \csc (e+f x)}} \]
((-12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]] - 2*Sin[2*(e + f*x)])/(10*f*Sqrt[d*Csc[e + f*x]])
Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 2030, 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.
\(\displaystyle \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc (e+f x)^2 \sqrt {d \csc (e+f x)}}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d^2 \int \frac {1}{(d \csc (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle d^2 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^2 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle d^2 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^2 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle d^2 \left (\frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 d^2 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
d^2*((-2*Cos[e + f*x])/(5*d*f*(d*Csc[e + f*x])^(3/2)) + (6*EllipticE[(e - Pi/2 + f*x)/2, 2])/(5*d^2*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]]))
3.6.24.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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 = 1.00 (sec) , antiderivative size = 442, normalized size of antiderivative = 6.14
method | result | size |
default | \(\frac {\sqrt {2}\, \left (-6 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+3 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+\sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-6 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-4 \sqrt {2}\, \cos \left (f x +e \right )+3 \sqrt {2}\right ) \csc \left (f x +e \right )}{5 f \sqrt {d \csc \left (f x +e \right )}}\) | \(442\) |
1/5/f*2^(1/2)*(-6*(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-c sc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*EllipticE((-I*(I-cot( f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)+3*(-I*(I-cot(f*x+e)+csc( f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f* x+e)))^(1/2)*EllipticF((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*c os(f*x+e)+2^(1/2)*cos(f*x+e)^3-6*(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I* (I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*Ellipt icE((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))+3*(-I*(I-cot(f*x+e)+ csc(f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+cs c(f*x+e)))^(1/2)*EllipticF((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2 ))-4*2^(1/2)*cos(f*x+e)+3*2^(1/2))/(d*csc(f*x+e))^(1/2)*csc(f*x+e)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {2 \, {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} + 3 \, \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{5 \, d f} \]
1/5*(2*(cos(f*x + e)^3 - cos(f*x + e))*sqrt(d/sin(f*x + e)) + 3*sqrt(2*I*d )*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*sqrt(-2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, c os(f*x + e) - I*sin(f*x + e))))/(d*f)
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \]