Integrand size = 21, antiderivative size = 90 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {3 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}+\frac {3 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{f} \]
-3*b^2*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f *x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2)-b*csc(f*x+e)*(b *sec(f*x+e))^(1/2)/f+3*b*sin(f*x+e)*(b*sec(f*x+e))^(1/2)/f
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\frac {b \sqrt {b \sec (e+f x)} \left (-\csc (e+f x)-3 \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+3 \sin (e+f x)\right )}{f} \]
(b*Sqrt[b*Sec[e + f*x]]*(-Csc[e + f*x] - 3*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + 3*Sin[e + f*x]))/f
Time = 0.44 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3105, 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.
\(\displaystyle \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^2 (b \sec (e+f x))^{3/2}dx\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {3}{2} \int (b \sec (e+f x))^{3/2}dx-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} \int \left (b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-b^2 \int \frac {1}{\sqrt {b \sec (e+f x)}}dx\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-b^2 \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {b^2 \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {b^2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {3}{2} \left (\frac {2 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{f}-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {b \csc (e+f x) \sqrt {b \sec (e+f x)}}{f}\) |
-((b*Csc[e + f*x]*Sqrt[b*Sec[e + f*x]])/f) + (3*((-2*b^2*EllipticE[(e + f* x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f))/2
3.4.95.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Simp[(-a)*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - 1))), x] + Simp[a^2*((m + n - 2)/(m - 1)) Int[(a*Csc[e + f* x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[ m, 1] && IntegersQ[2*m, 2*n] && !GtQ[n, 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 = 0.64 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.94
method | result | size |
default | \(-\frac {i b \sqrt {b \sec \left (f x +e \right )}\, \left (3 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )-3 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )+3 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )-3 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )-3 i \cot \left (f x +e \right )+2 i \csc \left (f x +e \right )\right )}{f}\) | \(265\) |
-I/f*b*(b*sec(f*x+e))^(1/2)*(3*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f *x+e)+1))^(1/2)*EllipticE(I*(-cot(f*x+e)+csc(f*x+e)),I)*cos(f*x+e)-3*(1/(c os(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-cot(f* x+e)+csc(f*x+e)),I)*cos(f*x+e)+3*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos (f*x+e)+1))^(1/2)*EllipticE(I*(-cot(f*x+e)+csc(f*x+e)),I)-3*(1/(cos(f*x+e) +1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-cot(f*x+e)+csc( f*x+e)),I)-3*I*cot(f*x+e)+2*I*csc(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.26 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\frac {-3 i \, \sqrt {2} b^{\frac {3}{2}} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {2} b^{\frac {3}{2}} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (3 \, b \cos \left (f x + e\right )^{2} - 2 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{2 \, f \sin \left (f x + e\right )} \]
1/2*(-3*I*sqrt(2)*b^(3/2)*sin(f*x + e)*weierstrassZeta(-4, 0, weierstrassP Inverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*I*sqrt(2)*b^(3/2)*sin(f *x + e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I *sin(f*x + e))) - 2*(3*b*cos(f*x + e)^2 - 2*b)*sqrt(b/cos(f*x + e)))/(f*si n(f*x + e))
Timed out. \[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^2} \,d x \]