Integrand size = 37, antiderivative size = 509 \[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=-\frac {\sqrt {d} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} b f}+\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}-\frac {2 \sqrt {2} a d \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \log \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+\sqrt {g} \tan (e+f x)\right )}{2 \sqrt {2} b f} \]
1/2*arctan(-1+2^(1/2)*d^(1/2)*(g*sin(f*x+e))^(1/2)/g^(1/2)/(d*cos(f*x+e))^ (1/2))*d^(1/2)*g^(1/2)/b/f*2^(1/2)+1/2*arctan(1+2^(1/2)*d^(1/2)*(g*sin(f*x +e))^(1/2)/g^(1/2)/(d*cos(f*x+e))^(1/2))*d^(1/2)*g^(1/2)/b/f*2^(1/2)+1/4*l n(g^(1/2)-2^(1/2)*d^(1/2)*(g*sin(f*x+e))^(1/2)/(d*cos(f*x+e))^(1/2)+g^(1/2 )*tan(f*x+e))*d^(1/2)*g^(1/2)/b/f*2^(1/2)-1/4*ln(g^(1/2)+2^(1/2)*d^(1/2)*( g*sin(f*x+e))^(1/2)/(d*cos(f*x+e))^(1/2)+g^(1/2)*tan(f*x+e))*d^(1/2)*g^(1/ 2)/b/f*2^(1/2)+2*a*d*EllipticPi((g*sin(f*x+e))^(1/2)/g^(1/2)/(1+cos(f*x+e) )^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*g^(1/2)*cos(f*x+e)^(1/2)/b/f/ (-a+b)^(1/2)/(a+b)^(1/2)/(d*cos(f*x+e))^(1/2)-2*a*d*EllipticPi((g*sin(f*x+ e))^(1/2)/g^(1/2)/(1+cos(f*x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2) *g^(1/2)*cos(f*x+e)^(1/2)/b/f/(-a+b)^(1/2)/(a+b)^(1/2)/(d*cos(f*x+e))^(1/2 )
Result contains complex when optimal does not.
Time = 11.40 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 \sqrt {2} g \sqrt {d \cos (e+f x)} \left (i \sqrt {-a-b} \sqrt {a-b} \operatorname {EllipticPi}\left (-i,\arcsin \left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right ),-1\right )-i \sqrt {-a-b} \sqrt {a-b} \operatorname {EllipticPi}\left (i,\arcsin \left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right ),-1\right )+a \left (-\operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {-a-b}},\arcsin \left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right ),-1\right )+\operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {-a-b}},\arcsin \left (\sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}\right ),-1\right )\right )\right ) \sqrt {\tan \left (\frac {1}{2} (e+f x)\right )}}{\sqrt {-a-b} \sqrt {a-b} b f \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {g \sin (e+f x)}} \]
(2*Sqrt[2]*g*Sqrt[d*Cos[e + f*x]]*(I*Sqrt[-a - b]*Sqrt[a - b]*EllipticPi[- I, ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] - I*Sqrt[-a - b]*Sqrt[a - b]*Ellipt icPi[I, ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] + a*(-EllipticPi[-(Sqrt[a - b] /Sqrt[-a - b]), ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] + EllipticPi[Sqrt[a - b]/Sqrt[-a - b], ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1]))*Sqrt[Tan[(e + f*x)/ 2]])/(Sqrt[-a - b]*Sqrt[a - b]*b*f*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*S qrt[g*Sin[e + f*x]])
Time = 1.57 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.94, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {3042, 3388, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3385, 3042, 3384, 993, 1542}
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.
| \(\displaystyle \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{a+b \sin \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle \frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {2 d^2 g \int \frac {g \tan (e+f x)}{d \left (\tan ^2(e+f x) g^2+g^2\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\int \frac {\tan (e+f x) g+g}{\tan ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}-\frac {\int \frac {g-g \tan (e+f x)}{\tan ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}+\frac {\int \frac {1}{\frac {\tan (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}}{2 d}-\frac {\int \frac {g-g \tan (e+f x)}{\tan ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{-\frac {g \tan (e+f x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {g \tan (e+f x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \tan (e+f x)}{\tan ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \tan (e+f x)}{\tan ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{\sqrt {d} \left (\frac {\tan (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{\sqrt {d} \left (\frac {\tan (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{\sqrt {d} \left (\frac {\tan (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}\right )}{\sqrt {d} \left (\frac {\tan (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{\frac {\tan (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 \sqrt {2} d \sqrt {g}}+\frac {\int \frac {\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{\frac {\tan (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \cos (e+f x)}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}}{2 d \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b}\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \cos (e+f x))}dx}{b \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{b \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {4 \sqrt {2} a d g \sqrt {\cos (e+f x)} \int \frac {g \sin (e+f x)}{(\cos (e+f x)+1) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}} \left (\frac {(a-b) \sin ^2(e+f x) g^2}{(\cos (e+f x)+1)^2}+(a+b) g^2\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{b f \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {4 \sqrt {2} a d g \sqrt {\cos (e+f x)} \left (\frac {\int \frac {1}{\left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \sin (e+f x)}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\left (\frac {\sqrt {b-a} \sin (e+f x) g}{\cos (e+f x)+1}+\sqrt {a+b} g\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{b f \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {d \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)}}+g \tan (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}-\frac {4 \sqrt {2} a d g \sqrt {\cos (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{b f \sqrt {d \cos (e+f x)}}\) |
(-4*Sqrt[2]*a*d*g*Sqrt[Cos[e + f*x]]*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[ a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1 ]/(Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b] , ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1]/(2*Sq rt[-a + b]*Sqrt[a + b]*Sqrt[g])))/(b*f*Sqrt[d*Cos[e + f*x]]) + (2*d^2*g*(( -(ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g])) + ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g* Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g]))/ (2*d) - (-1/2*Log[g - (Sqrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[g*Sin[e + f*x]])/Sqrt[ d*Cos[e + f*x]] + g*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[g + (Sqr t[2]*Sqrt[d]*Sqrt[g]*Sqrt[g*Sin[e + f*x]])/Sqrt[d*Cos[e + f*x]] + g*Tan[e + f*x]]/(2*Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*d)))/(b*f)
3.1.1.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f) Subst[Int[x^(k *(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2 , 0]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int[(g *Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a*(d/b) Int[(g*C os[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; F reeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (393 ) = 786\).
Time = 4.86 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.65
1/f*(-d*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1 ))^(1/2)*(g/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(1/ 2)*(I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a -I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b-I* EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a+I*Ell ipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b-Elliptic Pi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a+EllipticPi((- cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b-EllipticPi((-cot(f *x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a+EllipticPi((-cot(f*x+e) +csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b-(-a^2+b^2)^(1/2)*EllipticPi( (-cot(f*x+e)+csc(f*x+e)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1 /2))+a*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b ))^(1/2)),1/2*2^(1/2))-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),(a-b)/( a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b+(-a^2+b^2)^(1/2)*EllipticPi((-cot (f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2) )+a*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b) )^(1/2)),1/2*2^(1/2))-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/( -a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b)*(-csc(f*x+e)+cot(f*x+e))^(1/2)* (2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)/((1-c os(f*x+e))^2*csc(f*x+e)^2-1)/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^...
Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {d \cos {\left (e + f x \right )}} \sqrt {g \sin {\left (e + f x \right )}}}{a + b \cos {\left (e + f x \right )}}\, dx \]
\[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )} \sqrt {g \sin \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \cos \left (f x + e\right )} \sqrt {g \sin \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d \cos (e+f x)} \sqrt {g \sin (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {d\,\cos \left (e+f\,x\right )}\,\sqrt {g\,\sin \left (e+f\,x\right )}}{a+b\,\cos \left (e+f\,x\right )} \,d x \]