Integrand size = 23, antiderivative size = 103 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}} \]
a*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(1/2)+a*arctanh((e*sin(d*x+c))^ (1/2)/e^(1/2))/d/e^(1/2)-2*a*(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))*sin(d*x+c)^( 1/2)/d/(e*sin(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 0.95 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.19 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {9 a \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x)) \sin ^3(c+d x)}{5 d \left (4 \left (-2 \operatorname {AppellF1}\left (\frac {9}{4},\frac {1}{2},2,\frac {13}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )+\operatorname {AppellF1}\left (\frac {9}{4},\frac {3}{2},1,\frac {13}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right )+9 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right ) (-1+\cos (c+d x))\right ) \sqrt {e \sin (c+d x)}} \]
(9*a*AppellF1[5/4, 1/2, 1, 9/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2]*S ec[(c + d*x)/2]^2*(1 + Sec[c + d*x])*Sin[c + d*x]^3)/(5*d*(4*(-2*AppellF1[ 9/4, 1/2, 2, 13/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2] + AppellF1[9/4 , 3/2, 1, 13/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2])*Cos[(c + d*x)/2] ^2 + 9*AppellF1[5/4, 1/2, 1, 9/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2] *(-1 + Cos[c + d*x]))*Sqrt[e*Sin[c + d*x]])
Time = 0.61 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.94, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4360, 25, 25, 3042, 25, 3317, 25, 3042, 3044, 27, 266, 756, 216, 219, 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.
| \(\displaystyle \int \frac {a \sec (c+d x)+a}{\sqrt {e \sin (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sec (c+d x) (a (-\cos (c+d x))-a)}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {(\cos (c+d x) a+a) \sec (c+d x)}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sec (c+d x) (a \cos (c+d x)+a)}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\sqrt {e \cos \left (\frac {1}{2} (2 c-\pi )+d x\right )} \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-a \int -\frac {\sec (c+d x)}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+a \int \frac {\sec (c+d x)}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+a \int \frac {1}{\cos (c+d x) \sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \int \frac {e^2}{\sqrt {e \sin (c+d x)} \left (e^2-e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d e}+a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (e^2-e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d}+a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 a e \int \frac {1}{e^2-e^4 \sin ^4(c+d x)}d\sqrt {e \sin (c+d x)}}{d}+a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 a e \left (\frac {\int \frac {1}{e-e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e}+\frac {\int \frac {1}{e^2 \sin ^2(c+d x)+e}d\sqrt {e \sin (c+d x)}}{2 e}\right )}{d}+a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 a e \left (\frac {\int \frac {1}{e-e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e}+\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}\right )}{d}+a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+\frac {2 a e \left (\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}+\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}+\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}+\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 a e \left (\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}+\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 e^{3/2}}\right )}{d}+\frac {2 a \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}\) |
(2*a*e*(ArcTan[Sqrt[e]*Sin[c + d*x]]/(2*e^(3/2)) + ArcTanh[Sqrt[e]*Sin[c + d*x]]/(2*e^(3/2))))/d + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]])
3.2.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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[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[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
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 = 7.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.12
| method | result | size |
| parts | \(-\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a \left (\arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{\sqrt {e}\, d}\) | \(115\) |
| default | \(\frac {\frac {a \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}-\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(117\) |
-a*(-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))/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/d+a*(a rctan((e*sin(d*x+c))^(1/2)/e^(1/2))+arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))) /e^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 5.34 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\left [\frac {8 \, \sqrt {2} a \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, a \sqrt {-e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} - e \sin \left (d x + c\right ) - e\right )}}\right ) - a \sqrt {-e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right )}{8 \, d e}, \frac {8 \, \sqrt {2} a \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, a \sqrt {e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} + e \sin \left (d x + c\right ) - e\right )}}\right ) + a \sqrt {e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right )}{8 \, d e}\right ] \]
[1/8*(8*sqrt(2)*a*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*si n(d*x + c)) + 8*sqrt(2)*a*sqrt(I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*a*sqrt(-e)*arctan(1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*sqrt(e*sin(d*x + c))*sqrt(-e)/(e*cos(d*x + c)^2 - e*sin(d*x + c) - e)) - a*sqrt(-e)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(7*cos( d*x + c)^2 - (cos(d*x + c)^2 - 8)*sin(d*x + c) - 8)*sqrt(e*sin(d*x + c))*s qrt(-e) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)))/(d*e), 1/ 8*(8*sqrt(2)*a*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d *x + c)) + 8*sqrt(2)*a*sqrt(I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*a*sqrt(e)*arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e*sin(d*x + c))*sqrt(e)/(e*cos(d*x + c)^2 + e*sin(d*x + c) - e)) + a*sqrt(e)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(7*cos(d*x + c)^2 + (cos(d*x + c)^2 - 8)*sin(d*x + c) - 8)*sqrt(e*sin(d*x + c))*sqrt(e) - 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*co s(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)))/(d*e)]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=a \left (\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx\right ) \]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \]