Integrand size = 21, antiderivative size = 46 \[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:
-2*d^2*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 = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=-\frac {2 d^2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Input:
Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^2,x]
Output:
(-2*d^2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[ Sin[e + f*x]])
Time = 0.44 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 2030, 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.
\(\displaystyle \int \sin ^2(e+f x) (d \csc (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \csc (e+f x))^{3/2}}{\csc (e+f x)^2}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d^2 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
Input:
Int[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^2,x]
Output:
(2*d^2*EllipticE[(e - Pi/2 + f*x)/2, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[ e + f*x]])
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[(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 = 0.81 (sec) , antiderivative size = 238, normalized size of antiderivative = 5.17
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (\cos \left (f x +e \right )+1\right ) \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (-\cos \left (f x +e \right )-1\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-1+\cos \left (f x +e \right )\right )\right ) d \sqrt {d \csc \left (f x +e \right )}}{f}\) | \(238\) |
risch | \(-\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) \sqrt {2}\, d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}+\frac {\left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}\) | \(316\) |
Input:
int((d*csc(f*x+e))^(3/2)*sin(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
-1/f*2^(1/2)*(2*(1+I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(1-I*(-csc(f*x+e)+cot (f*x+e)))^(1/2)*(-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(cos(f*x+e)+1)*Ellipti cE((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2^(1/2))+(1+I*(-csc(f*x+e)+cot( f*x+e)))^(1/2)*(1-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(-I*(-csc(f*x+e)+cot(f *x+e)))^(1/2)*(-cos(f*x+e)-1)*EllipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2 ),1/2*2^(1/2))+2^(1/2)*(-1+cos(f*x+e)))*d*(d*csc(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33 \[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\frac {\sqrt {2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^2,x, algorithm="fricas")
Output:
(sqrt(2*I*d)*d*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e ) + I*sin(f*x + e))) + sqrt(-2*I*d)*d*weierstrassZeta(4, 0, weierstrassPIn verse(4, 0, cos(f*x + e) - I*sin(f*x + e))))/f
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\text {Timed out} \] Input:
integrate((d*csc(f*x+e))**(3/2)*sin(f*x+e)**2,x)
Output:
Timed out
\[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{2} \,d x } \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^2,x, algorithm="maxima")
Output:
integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^2, x)
\[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{2} \,d x } \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^2,x, algorithm="giac")
Output:
integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^2, x)
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \] Input:
int(sin(e + f*x)^2*(d/sin(e + f*x))^(3/2),x)
Output:
int(sin(e + f*x)^2*(d/sin(e + f*x))^(3/2), x)
\[ \int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \csc \left (f x +e \right ) \sin \left (f x +e \right )^{2}d x \right ) d \] Input:
int((d*csc(f*x+e))^(3/2)*sin(f*x+e)^2,x)
Output:
sqrt(d)*int(sqrt(csc(e + f*x))*csc(e + f*x)*sin(e + f*x)**2,x)*d