Integrand size = 25, antiderivative size = 101 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d \sqrt {e \sin (c+d x)}} \]
-2/3*e/a/d/(e*sin(d*x+c))^(3/2)+2/3*e*cos(d*x+c)/a/d/(e*sin(d*x+c))^(3/2)- 4/3*(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)*Elliptic F(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/a/d/(e*sin(d*x+c))^( 1/2)
Time = 1.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {2 \cot \left (\frac {1}{2} (c+d x)\right ) \left (-1+\cos (c+d x)-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{3 a d (1+\cos (c+d x)) \sqrt {e \sin (c+d x)}} \]
(2*Cot[(c + d*x)/2]*(-1 + Cos[c + d*x] - 2*EllipticF[(-2*c + Pi - 2*d*x)/4 , 2]*Sin[c + d*x]^(3/2)))/(3*a*d*(1 + Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])
Time = 0.74 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, 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, 25, 3318, 25, 3042, 3044, 15, 3047, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) \sqrt {e \sin (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cos (c+d x)}{(a (-\cos (c+d x))-a) \sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x)}{(\cos (c+d x) a+a) \sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cos (c+d x)}{(a \cos (c+d x)+a) \sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\sqrt {e \cos \left (\frac {1}{2} (2 c-\pi )+d x\right )} \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle -\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}}dx}{a}-\frac {e^2 \int -\frac {\cos (c+d x)}{(e \sin (c+d x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{5/2}}dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{5/2}}dx}{a}-\frac {e^2 \int \frac {\cos (c+d x)^2}{(e \sin (c+d x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {e \int \frac {1}{(e \sin (c+d x))^{5/2}}d(e \sin (c+d x))}{a d}-\frac {e^2 \int \frac {\cos (c+d x)^2}{(e \sin (c+d x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {e^2 \int \frac {\cos (c+d x)^2}{(e \sin (c+d x))^{5/2}}dx}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle -\frac {e^2 \left (-\frac {2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{3 e^2}-\frac {2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}\right )}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{3 e^2}-\frac {2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}\right )}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {e^2 \left (-\frac {2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 e^2 \sqrt {e \sin (c+d x)}}-\frac {2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}\right )}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 e^2 \sqrt {e \sin (c+d x)}}-\frac {2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}\right )}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {e^2 \left (-\frac {4 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}\right )}{a}-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}\) |
(-2*e)/(3*a*d*(e*Sin[c + d*x])^(3/2)) - (e^2*((-2*Cos[c + d*x])/(3*d*e*(e* Sin[c + d*x])^(3/2)) - (4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d* x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]])))/a
3.2.24.3.1 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[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 = 4.43 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {-\frac {2 e}{3 a \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {5}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\sin \left (d x +c \right )^{3}-\sin \left (d x +c \right )\right )}{3 a \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(121\) |
(-2/3/a*e/(e*sin(d*x+c))^(3/2)-2/3*((-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2) ^(1/2)*sin(d*x+c)^(5/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+sin(d *x+c)^3-sin(d*x+c))/a/sin(d*x+c)^2/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 = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - \sqrt {e \sin \left (d x + c\right )}\right )}}{3 \, {\left (a d e \cos \left (d x + c\right ) + a d e\right )}} \]
2/3*((sqrt(2)*cos(d*x + c) + sqrt(2))*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + (sqrt(2)*cos(d*x + c) + sqrt(2))*sqrt(I* e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - sqrt(e*sin(d *x + c)))/(a*d*e*cos(d*x + c) + a*d*e)
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \sin {\left (c + d x \right )}}}\, dx}{a} \]
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {e\,\sin \left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]