Integrand size = 12, antiderivative size = 71 \[ \int (d \csc (e+f x))^{3/2} \, dx=-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
-2*d*cos(f*x+e)*(d*csc(f*x+e))^(1/2)/f+2*d^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))/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int (d \csc (e+f x))^{3/2} \, dx=\frac {(d \csc (e+f x))^{3/2} \left (2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right ) \sin ^{\frac {3}{2}}(e+f x)-\sin (2 (e+f x))\right )}{f} \]
((d*Csc[e + f*x])^(3/2)*(2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x ]^(3/2) - Sin[2*(e + f*x)]))/f
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.
\(\displaystyle \int (d \csc (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d \csc (e+f x))^{3/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\) |
(-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]])
3.6.20.3.1 Defintions of rubi rules used
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 = 0.84 (sec) , antiderivative size = 413, normalized size of antiderivative = 5.82
method | result | size |
default | \(\frac {\sqrt {2}\, d \sqrt {d \csc \left (f x +e \right )}\, \left (2 \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \cos \left (f x +e \right )-\sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}-\sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right )}{f}\) | \(413\) |
1/f*2^(1/2)*d*(d*csc(f*x+e))^(1/2)*(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)*cos(f*x+e)-(I*( -I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(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*(-co t(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-cot(f*x+e)+csc(f *x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(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))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.17 \[ \int (d \csc (e+f x))^{3/2} \, dx=-\frac {2 \, d \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \]
-(2*d*sqrt(d/sin(f*x + e))*cos(f*x + e) + 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, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin( f*x + e))))/f
\[ \int (d \csc (e+f x))^{3/2} \, dx=\int \left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (d \csc (e+f x))^{3/2} \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
\[ \int (d \csc (e+f x))^{3/2} \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (d \csc (e+f x))^{3/2} \, dx=\int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \]