Integrand size = 25, antiderivative size = 139 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=-\frac {4 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d} \]
2/5*e*(e*sin(d*x+c))^(5/2)/a/d+4/21*e^4*(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))*s in(d*x+c)^(1/2)/a/d/(e*sin(d*x+c))^(1/2)-2/21*e^3*cos(d*x+c)*(e*sin(d*x+c) )^(1/2)/a/d+2/7*e^3*cos(d*x+c)^3*(e*sin(d*x+c))^(1/2)/a/d
Time = 1.74 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=\frac {e^3 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+(42+25 \cos (c+d x)-42 \cos (2 (c+d x))+15 \cos (3 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sqrt {e \sin (c+d x)}}{105 a d (1+\sec (c+d x)) \sqrt {\sin (c+d x)}} \]
(e^3*Cos[(c + d*x)/2]^2*Sec[c + d*x]*(40*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (42 + 25*Cos[c + d*x] - 42*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sq rt[Sin[c + d*x]])*Sqrt[e*Sin[c + d*x]])/(105*a*d*(1 + Sec[c + d*x])*Sqrt[S in[c + d*x]])
Time = 0.85 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4360, 25, 25, 3042, 3318, 3042, 3044, 15, 3048, 3042, 3049, 3042, 3121, 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.
| \(\displaystyle \int \frac {(e \sin (c+d x))^{7/2}}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2}}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{a (-\cos (c+d x))-a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{\cos (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{a \cos (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {e^2 \int \cos (c+d x) (e \sin (c+d x))^{3/2}dx}{a}-\frac {e^2 \int \cos ^2(c+d x) (e \sin (c+d x))^{3/2}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \int \cos (c+d x) (e \sin (c+d x))^{3/2}dx}{a}-\frac {e^2 \int \cos (c+d x)^2 (e \sin (c+d x))^{3/2}dx}{a}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {e \int (e \sin (c+d x))^{3/2}d(e \sin (c+d x))}{a d}-\frac {e^2 \int \cos (c+d x)^2 (e \sin (c+d x))^{3/2}dx}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \int \cos (c+d x)^2 (e \sin (c+d x))^{3/2}dx}{a}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \int \frac {\cos (c+d x)^2}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3049 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \left (\frac {2}{3} \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+\frac {2 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d e}\right )-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \left (\frac {2}{3} \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+\frac {2 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d e}\right )-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \left (\frac {2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d e}\right )-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \left (\frac {2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d e}\right )-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {e^2 \left (\frac {1}{7} e^2 \left (\frac {2 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {4 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {e \sin (c+d x)}}\right )-\frac {2 e \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 d}\right )}{a}\) |
(2*e*(e*Sin[c + d*x])^(5/2))/(5*a*d) - (e^2*((-2*e*Cos[c + d*x]^3*Sqrt[e*S in[c + d*x]])/(7*d) + (e^2*((4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[e*Sin[c + d*x] ])/(3*d*e)))/7))/a
3.2.20.3.1 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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
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]
Time = 5.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {\frac {2 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 a}+\frac {2 e^{4} \left (3 \sin \left (d x +c \right )^{5}+\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{21 a \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(128\) |
(2/5/a*e*(e*sin(d*x+c))^(5/2)+2/21*e^4*(3*sin(d*x+c)^5+(-sin(d*x+c)+1)^(1/ 2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2) ,1/2*2^(1/2))-5*sin(d*x+c)^3+2*sin(d*x+c))/a/cos(d*x+c)/(e*sin(d*x+c))^(1/ 2))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left (5 \, \sqrt {2} \sqrt {-i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (15 \, e^{3} \cos \left (d x + c\right )^{3} - 21 \, e^{3} \cos \left (d x + c\right )^{2} - 5 \, e^{3} \cos \left (d x + c\right ) + 21 \, e^{3}\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{105 \, a d} \]
-2/105*(5*sqrt(2)*sqrt(-I*e)*e^3*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*sqrt(I*e)*e^3*weierstrassPInverse(4, 0, cos(d* x + c) - I*sin(d*x + c)) - (15*e^3*cos(d*x + c)^3 - 21*e^3*cos(d*x + c)^2 - 5*e^3*cos(d*x + c) + 21*e^3)*sqrt(e*sin(d*x + c)))/(a*d)
Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]