Integrand size = 39, antiderivative size = 312 \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {g^2 (b+a \cos (e+f x)) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(a-b) c f \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {a+b \sec (e+f x)}}+\frac {2 g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {a+b \sec (e+f x)}}+\frac {g^2 (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{(a-b) f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}} \]
g^2*(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^2*(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^2*(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)+2*g^2*(cos(1/2*f*x+1/2*e )^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2,2^(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)
\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx \]
Time = 2.65 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 4467, 3042, 4346, 3042, 3286, 3042, 3284, 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.
| \(\displaystyle \int \frac {(g \sec (e+f x))^{5/2}}{(c \sec (e+f x)+c) \sqrt {a+b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (c \csc \left (e+f x+\frac {\pi }{2}\right )+c\right ) \sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4467 |
| \(\displaystyle \frac {g \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}}dx}{c}-g \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (\sec (e+f x) c+c)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{c}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {\sec (e+f x)}{\sqrt {b+a \cos (e+f x)}}dx}{c \sqrt {a+b \sec (e+f x)}}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {1}{\sin \left (e+f x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c \sqrt {a+b \sec (e+f x)}}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {\sec (e+f x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}}dx}{c \sqrt {a+b \sec (e+f x)}}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {1}{\sin \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (e+f x+\frac {\pi }{2}\right )}{a+b}}}dx}{c \sqrt {a+b \sec (e+f x)}}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-g \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (\csc \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx\) |
| \(\Big \downarrow \) 4463 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {1}{\sqrt {b+a \cos (e+f x)} (\cos (e+f x) c+c)}dx}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (\sin \left (e+f x+\frac {\pi }{2}\right ) c+c\right )}dx}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3247 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (-\frac {a \int -\frac {\cos (e+f x) c+c}{2 \sqrt {b+a \cos (e+f x)}}dx}{c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \int \frac {\cos (e+f x) c+c}{\sqrt {b+a \cos (e+f x)}}dx}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right ) c+c}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \cos (e+f x)}}dx}{a}+\frac {c \int \sqrt {b+a \cos (e+f x)}dx}{a}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \int \sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{a}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \sqrt {a \cos (e+f x)+b} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}dx}{a \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {c \sqrt {a \cos (e+f x)+b} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (e+f x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}}dx}{a \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (e+f x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g^2 \sqrt {g \sec (e+f x)} \sqrt {a \cos (e+f x)+b} \left (\frac {a \left (\frac {2 c (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{a f \sqrt {a \cos (e+f x)+b}}+\frac {2 c \sqrt {a \cos (e+f x)+b} E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{a f \sqrt {\frac {a \cos (e+f x)+b}{a+b}}}\right )}{2 c^2 (a-b)}-\frac {\sin (e+f x) \sqrt {a \cos (e+f x)+b}}{f (a-b) (c \cos (e+f x)+c)}\right )}{\sqrt {a+b \sec (e+f x)}}\) |
(2*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (e + f*x)/2, (2*a) /(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g^2*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)]*EllipticF[(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]]
3.3.78.3.1 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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 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]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(5/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[g/d Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[c*(g/d) Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[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 = 7.97 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(\frac {i \left (4 \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) a -2 b \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-a \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-4 \operatorname {EllipticPi}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {a -b}{a +b}}\right ) a +4 b \operatorname {EllipticPi}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {g \sec \left (f x +e \right )}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, g^{2} \cos \left (f x +e \right )}{c f \left (a -b \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \left (b +a \cos \left (f x +e \right )\right )}\) | \(323\) |
I/c/f/(a-b)*(4*EllipticF(I*(-cot(f*x+e)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2))* a-2*b*EllipticF(I*(-cot(f*x+e)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2))-a*Ellipti cE(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))-4*EllipticPi(I*(-cot(f*x+e)+csc(f*x +e)),-1,I*((a-b)/(a+b))^(1/2))*a+4*b*EllipticPi(I*(-cot(f*x+e)+csc(f*x+e)) ,-1,I*((a-b)/(a+b))^(1/2)))*(a+b*sec(f*x+e))^(1/2)*(g*sec(f*x+e))^(1/2)*(1 /(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*g^2/(1/(cos(f*x+e)+1))^(1/2) /(b+a*cos(f*x+e))*cos(f*x+e)
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
\[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
Timed out. \[ \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]