Integrand size = 39, antiderivative size = 229 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {g (b+a \cos (e+f x)) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(a-b) c f \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{(a-b) f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}} \]
-g*(b+a*cos(f*x+e))*sin(f*x+e)*(g*sec(f*x+e))^(1/2)/(a-b)/f/(c+c*cos(f*x+e ))/(a+b*sec(f*x+e))^(1/2)+g*(b+a*cos(f*x+e))*(cos(1/2*f*x+1/2*e)^2)^(1/2)/ cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2)*(a/(a+b))^(1/2))*( g*sec(f*x+e))^(1/2)/(a-b)/c/f/((b+a*cos(f*x+e))/(a+b))^(1/2)/(a+b*sec(f*x+ e))^(1/2)+g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin( 1/2*f*x+1/2*e),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(f*x+e))/(a+b))^(1/2)*(g* sec(f*x+e))^(1/2)/c/f/(a+b*sec(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 8.39 (sec) , antiderivative size = 1019, normalized size of antiderivative = 4.45 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (b+a \cos (e+f x)) (g \sec (e+f x))^{3/2} \left (\frac {2 \csc (e)}{(-a+b) f}+\frac {2 \sec \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}+\frac {f x}{2}\right ) \sin \left (\frac {f x}{2}\right )}{(-a+b) f}\right )}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\csc (e) \left (b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))\right )}{a \sqrt {1+\cot ^2(e)} \left (1+\frac {b \csc (e)}{a \sqrt {1+\cot ^2(e)}}\right )},\frac {\csc (e) \left (b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))\right )}{a \sqrt {1+\cot ^2(e)} \left (-1+\frac {b \csc (e)}{a \sqrt {1+\cot ^2(e)}}\right )}\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sqrt {b+a \cos (e+f x)} \csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) (g \sec (e+f x))^{3/2} \sec (f x-\arctan (\cot (e))) \sqrt {\frac {a \sqrt {1+\cot ^2(e)}-a \sqrt {1+\cot ^2(e)} \sin (f x-\arctan (\cot (e)))}{a \sqrt {1+\cot ^2(e)}-b \csc (e)}} \sqrt {\frac {a \sqrt {1+\cot ^2(e)}+a \sqrt {1+\cot ^2(e)} \sin (f x-\arctan (\cot (e)))}{a \sqrt {1+\cot ^2(e)}+b \csc (e)}} \sqrt {b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))}}{(-a+b) f \sqrt {1+\cot ^2(e)} \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}+\frac {a \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sqrt {b+a \cos (e+f x)} \csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) (g \sec (e+f x))^{3/2} \left (\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {\sec (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a \sqrt {1+\tan ^2(e)} \left (1-\frac {b \sec (e)}{a \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a \sqrt {1+\tan ^2(e)} \left (-1-\frac {b \sec (e)}{a \sqrt {1+\tan ^2(e)}}\right )}\right ) \sin (f x+\arctan (\tan (e))) \tan (e)}{\sqrt {1+\tan ^2(e)} \sqrt {\frac {a \sqrt {1+\tan ^2(e)}-a \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{b \sec (e)+a \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {a \sqrt {1+\tan ^2(e)}+a \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{-b \sec (e)+a \sqrt {1+\tan ^2(e)}}} \sqrt {b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}-\frac {\frac {\sin (f x+\arctan (\tan (e))) \tan (e)}{\sqrt {1+\tan ^2(e)}}+\frac {2 a \cos (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a^2 \cos ^2(e)+a^2 \sin ^2(e)}}{\sqrt {b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}\right )}{2 (-a+b) f \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \]
(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*(g*Sec[e + f*x])^(3/2)*((2*Csc[ e])/((-a + b)*f) + (2*Sec[e/2]*Sec[e/2 + (f*x)/2]*Sin[(f*x)/2])/((-a + b)* f)))/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e ]]]))/(a*Sqrt[1 + Cot[e]^2]*(1 + (b*Csc[e])/(a*Sqrt[1 + Cot[e]^2]))), (Csc [e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]))/(a*Sqrt[1 + Cot[e]^2]*(-1 + (b*Csc[e])/(a*Sqrt[1 + Cot[e]^2])))]*Cos[e/2 + (f*x)/2] ^2*Sqrt[b + a*Cos[e + f*x]]*Csc[e/2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*Sec[f *x - ArcTan[Cot[e]]]*Sqrt[(a*Sqrt[1 + Cot[e]^2] - a*Sqrt[1 + Cot[e]^2]*Sin [f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] - b*Csc[e])]*Sqrt[(a*Sqrt[1 + Cot[e]^2] + a*Sqrt[1 + Cot[e]^2]*Sin[f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] + b*Csc[e])]*Sqrt[b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcT an[Cot[e]]]])/((-a + b)*f*Sqrt[1 + Cot[e]^2]*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (a*Cos[e/2 + (f*x)/2]^2*Sqrt[b + a*Cos[e + f*x]]*Csc[e /2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((S ec[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt [1 + Tan[e]^2]*(1 - (b*Sec[e])/(a*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(b + a *Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt[1 + Tan[e]^ 2]*(-1 - (b*Sec[e])/(a*Sqrt[1 + Tan[e]^2]))))]*Sin[f*x + ArcTan[Tan[e]]]*T an[e])/(Sqrt[1 + Tan[e]^2]*Sqrt[(a*Sqrt[1 + Tan[e]^2] - a*Cos[f*x + Arc...
Time = 1.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.359, Rules used = {3042, 4463, 3042, 3247, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {(g \sec (e+f x))^{3/2}}{(c \sec (e+f x)+c) \sqrt {a+b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (c \csc \left (e+f x+\frac {\pi }{2}\right )+c\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4463 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {1}{\sqrt {b+a \cos (e+f x)} (\cos (e+f x) c+c)}dx}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (\sin \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3247 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (-\frac {a \int -\frac {\cos (e+f x) c+c}{2 \sqrt {b+a \cos (e+f x)}}dx}{c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \int \frac {\cos (e+f x) c+c}{\sqrt {b+a \cos (e+f x)}}dx}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right ) c+c}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \cos (e+f x)}}dx}{a}+\frac {c \int \sqrt {b+a \cos (e+f x)}dx}{a}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \int \sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{a}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \sqrt {a \cos (e+f x)+b} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}dx}{a \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \sqrt {a \cos (e+f x)+b} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (e+f x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}}dx}{a \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (e+f x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {2 c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{a f \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
(g*Sqrt[b + a*Cos[e + f*x]]*Sqrt[g*Sec[e + f*x]]*((a*((2*c*Sqrt[b + a*Cos[ e + f*x]]*EllipticE[(e + f*x)/2, (2*a)/(a + b)])/(a*f*Sqrt[(b + a*Cos[e + f*x])/(a + b)]) + (2*(a - b)*c*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Elliptic F[(e + f*x)/2, (2*a)/(a + b)])/(a*f*Sqrt[b + a*Cos[e + f*x]])))/(2*(a - b) *c^2) - (Sqrt[b + a*Cos[e + f*x]]*Sin[e + f*x])/((a - b)*f*(c + c*Cos[e + f*x]))))/Sqrt[a + b*Sec[e + f*x]]
3.3.77.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[g*Sqr t[g*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]) Int [1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 4.70 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {i g \cos \left (f x +e \right ) \sqrt {g \sec \left (f x +e \right )}\, \sqrt {a +b \sec \left (f x +e \right )}\, \left (2 \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) a -a \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}}{c f \left (a -b \right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(207\) |
-I*g/c/f/(a-b)*cos(f*x+e)*(g*sec(f*x+e))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(2*E llipticF(I*(-cot(f*x+e)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2))*a-a*EllipticE(I* (-cot(f*x+e)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2))-b*EllipticE(I*(-cot(f*x+e)+ csc(f*x+e)),(-(a-b)/(a+b))^(1/2)))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1 ))^(1/2)/(b+a*cos(f*x+e))/(1/(cos(f*x+e)+1))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.26 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {6 \, a g \sqrt {\frac {a \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, {\left (3 \, a - 2 \, b\right )} g \cos \left (f x + e\right ) + i \, {\left (3 \, a - 2 \, b\right )} g\right )} \sqrt {a g} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) + 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (-i \, {\left (3 \, a - 2 \, b\right )} g \cos \left (f x + e\right ) - i \, {\left (3 \, a - 2 \, b\right )} g\right )} \sqrt {a g} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) - 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (i \, a g \cos \left (f x + e\right ) + i \, a g\right )} \sqrt {a g} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) + 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (-i \, a g \cos \left (f x + e\right ) - i \, a g\right )} \sqrt {a g} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) - 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right )\right )}{6 \, {\left ({\left (a^{2} - a b\right )} c f \cos \left (f x + e\right ) + {\left (a^{2} - a b\right )} c f\right )}} \]
-1/6*(6*a*g*sqrt((a*cos(f*x + e) + b)/cos(f*x + e))*sqrt(g/cos(f*x + e))*c os(f*x + e)*sin(f*x + e) + sqrt(2)*(I*(3*a - 2*b)*g*cos(f*x + e) + I*(3*a - 2*b)*g)*sqrt(a*g)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9* a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) + 3*I*a*sin(f*x + e) + 2*b)/a) + sqrt(2)*(-I*(3*a - 2*b)*g*cos(f*x + e) - I*(3*a - 2*b)*g)*sqrt(a*g)*weier strassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*( 3*a*cos(f*x + e) - 3*I*a*sin(f*x + e) + 2*b)/a) - 3*sqrt(2)*(I*a*g*cos(f*x + e) + I*a*g)*sqrt(a*g)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9 *a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9 *a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) + 3*I*a*sin(f*x + e) + 2*b)/a)) - 3*sqrt(2)*(-I*a*g*cos(f*x + e) - I*a*g)*sqrt(a*g)*weierstrassZeta(-4/3* (3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3* (3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) - 3 *I*a*sin(f*x + e) + 2*b)/a)))/((a^2 - a*b)*c*f*cos(f*x + e) + (a^2 - a*b)* c*f)
\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]
Integral((g*sec(e + f*x))**(3/2)/(sqrt(a + b*sec(e + f*x))*sec(e + f*x) + sqrt(a + b*sec(e + f*x))), x)/c
\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]