Integrand size = 21, antiderivative size = 103 \[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=-\frac {2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 d^2 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 d \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{21 f} \]
-2/7*d^4*cos(f*x+e)/f/(d*csc(f*x+e))^(5/2)-10/21*d^2*cos(f*x+e)/f/(d*csc(f *x+e))^(1/2)-10/21*d*(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)/f
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66 \[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=-\frac {d \sqrt {d \csc (e+f x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+26 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 f} \]
-1/84*(d*Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt [Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/f
Time = 0.49 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 2030, 4256, 3042, 4256, 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.
\(\displaystyle \int \sin ^5(e+f x) (d \csc (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \csc (e+f x))^{3/2}}{\csc (e+f x)^5}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d^5 \int \frac {1}{(d \csc (e+f x))^{7/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle d^5 \left (\frac {5 \int \frac {1}{(d \csc (e+f x))^{3/2}}dx}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^5 \left (\frac {5 \int \frac {1}{(d \csc (e+f x))^{3/2}}dx}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle d^5 \left (\frac {5 \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^5 \left (\frac {5 \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle d^5 \left (\frac {5 \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^5 \left (\frac {5 \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle d^5 \left (\frac {5 \left (\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\) |
d^5*((-2*Cos[e + f*x])/(7*d*f*(d*Csc[e + f*x])^(5/2)) + (5*((-2*Cos[e + f* x])/(3*d*f*Sqrt[d*Csc[e + f*x]]) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(3*d^2*f)))/(7*d^2))
3.6.15.3.1 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[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 = 1.26 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {\sqrt {2}\, \left (5 i \sin \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-3 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}+3 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )+8 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-8 \sqrt {2}\, \cos \left (f x +e \right )\right ) d \csc \left (f x +e \right ) \sqrt {d \csc \left (f x +e \right )}\, \left (\cos \left (f x +e \right )+1\right )}{21 f}\) | \(186\) |
1/21/f*2^(1/2)*(5*I*sin(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)*Elliptic F((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))-3*cos(f*x+e)^4*2^(1/2) +3*2^(1/2)*cos(f*x+e)^3+8*cos(f*x+e)^2*2^(1/2)-8*2^(1/2)*cos(f*x+e))*d*csc (f*x+e)*(d*csc(f*x+e))^(1/2)*(cos(f*x+e)+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 = 0.96 \[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=\frac {2 \, {\left (3 \, d \cos \left (f x + e\right )^{3} - 8 \, d \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - 5 i \, \sqrt {2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {-2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{21 \, f} \]
1/21*(2*(3*d*cos(f*x + e)^3 - 8*d*cos(f*x + e))*sqrt(d/sin(f*x + e))*sin(f *x + e) - 5*I*sqrt(2*I*d)*d*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin (f*x + e)) + 5*I*sqrt(-2*I*d)*d*weierstrassPInverse(4, 0, cos(f*x + e) - I *sin(f*x + e)))/f
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=\text {Timed out} \]
\[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{5} \,d x } \]
\[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{5} \,d x } \]
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^5\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \]