Integrand size = 21, antiderivative size = 77 \[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 d^2 f} \]
-2/3*cos(f*x+e)*(d*csc(f*x+e))^(3/2)/d^3/f-2/3*(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)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^( 1/2))*(d*csc(f*x+e))^(1/2)*sin(f*x+e)^(1/2)/d^2/f
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \csc ^3(e+f x) \left (\cos (e+f x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 f (d \csc (e+f x))^{3/2}} \]
(-2*Csc[e + f*x]^3*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin [e + f*x]^(3/2)))/(3*f*(d*Csc[e + f*x])^(3/2))
Time = 0.34 (sec) , antiderivative size = 79, 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, 3120}
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.
\(\displaystyle \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{5/2}dx}{d^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{5/2}dx}{d^4}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\frac {1}{3} d^2 \int \sqrt {d \csc (e+f x)}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{3} d^2 \int \sqrt {d \csc (e+f x)}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^4}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {1}{3} d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{3} d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^4}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {2 d^2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^4}\) |
((-2*d*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*f) + (2*d^2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(3*f))/d^4
3.6.36.3.1 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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 = 0.96 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.30
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )+i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right )-\sqrt {2}\, \cos \left (f x +e \right )\right )}{3 f d \sqrt {d \csc \left (f x +e \right )}\, \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(254\) |
-1/3/f*2^(1/2)*(I*sin(f*x+e)*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))+I*(-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))*sin(f*x+e)-2^(1/2)*cos(f*x+e))/d/(d*csc(f*x+e))^(1/2)/(cos(f*x+e )^2-1)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, d^{2} f \sin \left (f x + e\right )} \]
1/3*(-I*sqrt(2*I*d)*sin(f*x + e)*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(-2*I*d)*sin(f*x + e)*weierstrassPInverse(4, 0, co s(f*x + e) - I*sin(f*x + e)) - 2*sqrt(d/sin(f*x + e))*cos(f*x + e))/(d^2*f *sin(f*x + e))
\[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {\csc ^{4}{\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{4}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^4\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]