Integrand size = 25, antiderivative size = 149 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=-\frac {2 \cos (c+d x)}{21 a d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a d e^3 \sqrt {e \csc (c+d x)}}-\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{21 a d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \sin ^2(c+d x)}{5 a d e^3 \sqrt {e \csc (c+d x)}} \]
-2/21*cos(d*x+c)/a/d/e^3/(e*csc(d*x+c))^(1/2)+2/7*cos(d*x+c)^3/a/d/e^3/(e* csc(d*x+c))^(1/2)+2/5*sin(d*x+c)^2/a/d/e^3/(e*csc(d*x+c))^(1/2)+4/21*(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)*EllipticF(cos(1/2 *c+1/4*Pi+1/2*d*x),2^(1/2))/a/d/e^3/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)
Time = 1.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\frac {\sqrt {e \csc (c+d x)} \left (80 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}+126 \sin (c+d x)+10 \sin (2 (c+d x))-42 \sin (3 (c+d x))+15 \sin (4 (c+d x))\right )}{420 a d e^4} \]
(Sqrt[e*Csc[c + d*x]]*(80*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + 126*Sin[c + d*x] + 10*Sin[2*(c + d*x)] - 42*Sin[3*(c + d*x)] + 15 *Sin[4*(c + d*x)]))/(420*a*d*e^4)
Time = 0.79 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3048, 3042, 3049, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) (e \csc (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int \frac {\sin ^{\frac {7}{2}}(c+d x)}{\sec (c+d x) a+a}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^{7/2}}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int -\frac {\cos (c+d x) \sin ^{\frac {7}{2}}(c+d x)}{-\cos (c+d x) a-a}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {\cos (c+d x) \sin ^{\frac {7}{2}}(c+d x)}{\cos (c+d x) a+a}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \sin ^{\frac {7}{2}}(c+d x)}{\cos (c+d x) a+a}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2} \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\frac {\int \cos (c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \cos (c+d x) \sin (c+d x)^{3/2}dx}{a}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^{3/2}dx}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\frac {\int \sin ^{\frac {3}{2}}(c+d x)d\sin (c+d x)}{a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^{3/2}dx}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^{3/2}dx}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\frac {1}{7} \int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\frac {1}{7} \int \frac {\cos (c+d x)^2}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3049 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\frac {1}{7} \left (\frac {2}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx+\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )-\frac {2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\frac {1}{7} \left (\frac {2}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx+\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )-\frac {2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 \sin ^{\frac {5}{2}}(c+d x)}{5 a d}-\frac {\frac {1}{7} \left (\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}+\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )-\frac {2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}}{a}}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
(-((((4*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) + (2*Cos[c + d*x]*Sqrt[Sin [c + d*x]])/(3*d))/7 - (2*Cos[c + d*x]^3*Sqrt[Sin[c + d*x]])/(7*d))/a) + ( 2*Sin[c + d*x]^(5/2))/(5*a*d))/(e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x] ])
3.3.99.3.1 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_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(b*Sin[e + f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/ (b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Sin[e + f*x])^n*(a *Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 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 = 10.01 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-10 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \cos \left (d x +c \right )+15 \sqrt {2}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )-10 i \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}-21 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sin \left (d x +c \right )-5 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}+21 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )^{3}}{105 a d \,e^{3} \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\) | \(316\) |
1/105/a/d*2^(1/2)*(-10*I*(I*(-I+cot(d*x+c)-csc(d*x+c)))^(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))*cos(d*x+c)+15*2^(1/2)*cos(d* x+c)^3*sin(d*x+c)-10*I*(I*(-I+cot(d*x+c)-csc(d*x+c)))^(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))-21*cos(d*x+c)^2*2^(1/2)*sin(d* x+c)-5*cos(d*x+c)*sin(d*x+c)*2^(1/2)+21*2^(1/2)*sin(d*x+c))/e^3/(e*csc(d*x +c))^(1/2)/(cos(d*x+c)-1)^2/(cos(d*x+c)+1)^2*sin(d*x+c)^3
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\frac {2 \, {\left ({\left (15 \, \cos \left (d x + c\right )^{3} - 21 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 21\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 5 i \, \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{105 \, a d e^{4}} \]
2/105*((15*cos(d*x + c)^3 - 21*cos(d*x + c)^2 - 5*cos(d*x + c) + 21)*sqrt( e/sin(d*x + c))*sin(d*x + c) + 5*I*sqrt(2*I*e)*weierstrassPInverse(4, 0, c os(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(-2*I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)))/(a*d*e^4)
Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{7/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]