Integrand size = 25, antiderivative size = 155 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a d}+\frac {2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}-\frac {2 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a d}+\frac {4 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a d} \]
-4/21*e^2*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a/d+2/7*e^2*cot(d*x+c)*csc(d*x+c )^2*(e*csc(d*x+c))^(1/2)/a/d-2/7*e^2*csc(d*x+c)^3*(e*csc(d*x+c))^(1/2)/a/d -4/21*e^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*El lipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c) ^(1/2)/a/d
Time = 1.79 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{5/2} \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left ((2+\cos (c+d x)-2 \cos (2 (c+d x))-\cos (3 (c+d x))) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+2 (4+2 \cos (c+d x)+\cos (2 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sin ^{\frac {5}{2}}(c+d x)}{168 a d} \]
-1/168*(Csc[(c + d*x)/2]^2*(e*Csc[c + d*x])^(5/2)*Sec[(c + d*x)/2]^4*((2 + Cos[c + d*x] - 2*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*EllipticF[(-2*c + P i - 2*d*x)/4, 2] + 2*(4 + 2*Cos[c + d*x] + Cos[2*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sin[c + d*x]^(5/2))/(a*d)
Time = 0.79 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.82, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 25, 3318, 25, 3042, 3044, 15, 3047, 3042, 3116, 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.
| \(\displaystyle \int \frac {(e \csc (c+d x))^{5/2}}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{(\sec (c+d x) a+a) \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {\cos (c+d x)}{(-\cos (c+d x) a-a) \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {\cos (c+d x)}{(\cos (c+d x) a+a) \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos (c+d x)}{(\cos (c+d x) a+a) \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^{5/2} \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}dx}{a}+\frac {\int -\frac {\cos (c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}dx}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}dx}{a}-\frac {\int \frac {\cos (c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}dx}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{9/2}}dx}{a}-\frac {\int \frac {\cos (c+d x)}{\sin (c+d x)^{9/2}}dx}{a}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{9/2}}dx}{a}-\frac {\int \frac {1}{\sin ^{\frac {9}{2}}(c+d x)}d\sin (c+d x)}{a d}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {\int \frac {\cos (c+d x)^2}{\sin (c+d x)^{9/2}}dx}{a}+\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {-\frac {2}{7} \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)}dx-\frac {2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}}{a}+\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {-\frac {2}{7} \int \frac {1}{\sin (c+d x)^{5/2}}dx-\frac {2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}}{a}+\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {-\frac {2}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}}{a}+\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {-\frac {2}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}}{a}+\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\frac {2}{7 a d \sin ^{\frac {7}{2}}(c+d x)}+\frac {-\frac {2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )}{a}\right )\) |
-(e^2*Sqrt[e*Csc[c + d*x]]*(((-2*((2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3* d) - (2*Cos[c + d*x])/(3*d*Sin[c + d*x]^(3/2))))/7 - (2*Cos[c + d*x])/(7*d *Sin[c + d*x]^(7/2)))/a + 2/(7*a*d*Sin[c + d*x]^(7/2)))*Sqrt[Sin[c + d*x]] )
3.3.93.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ (b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Cos[e + f*x] )^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 10.68 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {\sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (8 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-3 \left (1-\cos \left (d x +c \right )\right )^{6} \csc \left (d x +c \right )^{6}-5 \left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-9 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-7\right ) \csc \left (d x +c \right )^{2}}{84 a d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\) | \(360\) |
1/84/a/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+c) ))^(5/2)*(1-cos(d*x+c))^2*(8*I*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*2^(1/2 )*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-cot(d*x+c)+csc(d*x+c)))^(1/2)* EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*(-cot(d*x+c)+c sc(d*x+c))-3*(1-cos(d*x+c))^6*csc(d*x+c)^6-5*(1-cos(d*x+c))^4*csc(d*x+c)^4 -9*(1-cos(d*x+c))^2*csc(d*x+c)^2-7)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^2/(( 1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)*csc(d*x+c))^(1/2)/((1-cos( d*x+c))^3*csc(d*x+c)^3+csc(d*x+c)-cot(d*x+c))^(1/2)*csc(d*x+c)^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.05 \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left ({\left (i \, e^{2} \cos \left (d x + c\right ) + i \, e^{2}\right )} \sqrt {2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (-i \, e^{2} \cos \left (d x + c\right ) - i \, e^{2}\right )} \sqrt {-2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (2 \, e^{2} \cos \left (d x + c\right )^{2} + 2 \, e^{2} \cos \left (d x + c\right ) + 3 \, e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{21 \, {\left (a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]
-2/21*((I*e^2*cos(d*x + c) + I*e^2)*sqrt(2*I*e)*sin(d*x + c)*weierstrassPI nverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + (-I*e^2*cos(d*x + c) - I*e^2 )*sqrt(-2*I*e)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin (d*x + c)) + (2*e^2*cos(d*x + c)^2 + 2*e^2*cos(d*x + c) + 3*e^2)*sqrt(e/si n(d*x + c)))/((a*d*cos(d*x + c) + a*d)*sin(d*x + c))
Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]