Integrand size = 21, antiderivative size = 73 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
-2*cos(f*x+e)*(d*csc(f*x+e))^(1/2)/d^2/f+2*(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) )/d/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {-2 \cot (e+f x)+\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}}{d f \sqrt {d \csc (e+f x)}} \]
(-2*Cot[e + f*x] + (2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f* x]])/(d*f*Sqrt[d*Csc[e + f*x]])
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2030, 3042, 4255, 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.
\(\displaystyle \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{3/2}dx}{d^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{3/2}dx}{d^3}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^3}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^3}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {-\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^3}\) |
((-2*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/f - (2*d^2*EllipticE[(e - Pi/2 + f*x)/2, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]]))/d^3
3.6.35.3.1 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[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 = 0.92 (sec) , antiderivative size = 421, normalized size of antiderivative = 5.77
method | result | size |
default | \(\frac {\sqrt {2}\, \left (2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \csc \left (f x +e \right )}{f \sqrt {d \csc \left (f x +e \right )}\, d}\) | \(421\) |
1/f*2^(1/2)*(2*(-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)*EllipticE((-I*(I-cot(f*x +e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)-(-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))*cos(f* x+e)+2*(-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)*EllipticE((-I*(I-cot(f*x+e)+csc( f*x+e)))^(1/2),1/2*2^(1/2))-(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(I+co t(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))-2^(1/2))/(d*csc(f*x+e))^( 1/2)/d*csc(f*x+e)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d^{2} f} \]
-(2*sqrt(d/sin(f*x + e))*cos(f*x + e) + sqrt(2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + sqrt(-2*I*d)*w eierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))))/(d^2*f)
\[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{3}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{3}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^3\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]