Integrand size = 37, antiderivative size = 616 \[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {a d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f \sqrt {g}}-\frac {a d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^2 f \sqrt {g}}-\frac {2 \sqrt {2} a^2 d^{5/2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 \sqrt {-a^2+b^2} f \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {2} a^2 d^{5/2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^2 \sqrt {-a^2+b^2} f \sqrt {g \cos (e+f x)}}-\frac {a d^{5/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f \sqrt {g}}+\frac {a d^{5/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b^2 f \sqrt {g}}-\frac {d^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b f g}+\frac {d^3 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
1/2*a*d^(5/2)*arctan(1-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos (f*x+e))^(1/2))/b^2/f*2^(1/2)/g^(1/2)-1/2*a*d^(5/2)*arctan(1+2^(1/2)*g^(1/ 2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f*x+e))^(1/2))/b^2/f*2^(1/2)/g^(1/2 )-1/4*a*d^(5/2)*ln(d^(1/2)-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x +e))^(1/2)+d^(1/2)*tan(f*x+e))/b^2/f*2^(1/2)/g^(1/2)+1/4*a*d^(5/2)*ln(d^(1 /2)+2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2)+d^(1/2)*tan( f*x+e))/b^2/f*2^(1/2)/g^(1/2)-2*a^2*d^(5/2)*EllipticPi((d*sin(f*x+e))^(1/2 )/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b-(-a^2+b^2)^(1/2)),I)*2^(1/2)*cos(f*x+ e)^(1/2)/b^2/f/(-a^2+b^2)^(1/2)/(g*cos(f*x+e))^(1/2)+2*a^2*d^(5/2)*Ellipti cPi((d*sin(f*x+e))^(1/2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/ 2)),I)*2^(1/2)*cos(f*x+e)^(1/2)/b^2/f/(-a^2+b^2)^(1/2)/(g*cos(f*x+e))^(1/2 )-d^2*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/b/f/g-1/2*d^3*(sin(e+1/4*P i+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*sin (2*f*x+2*e)^(1/2)/b/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 25.01 (sec) , antiderivative size = 1318, normalized size of antiderivative = 2.14 \[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \]
(Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)*((2*Sqrt[Sin[e + f*x]]*((Sqrt[a ]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Lo g[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b ^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - ( b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2 )/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f *x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Sin[e + f*x]]*(b*Tan[e + f *x] + a*Sqrt[1 + Tan[e + f*x]^2])*(-20*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[T an[e + f*x]]] + 20*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + (10* Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[T an[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(3/4) - (10*Sqrt[2]*Sqrt[a]*(2*a^2 - b ^2)*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a ^2 - b^2)^(3/4) + 10*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 10*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1 /4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(3/4) + (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^...
Time = 2.61 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.622, Rules used = {3042, 3388, 3042, 3048, 3042, 3053, 3042, 3120, 3388, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3387, 3042, 3386, 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.
| \(\displaystyle \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle \frac {d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \int \frac {d \tan (e+f x)}{g \left (\tan ^2(e+f x) d^2+d^2\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}+\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} g}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {d} g}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3387 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{b \sqrt {g \cos (e+f x)}}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{b \sqrt {g \cos (e+f x)}}\right )}{b}\) |
| \(\Big \downarrow \) 3386 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} d \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}+\frac {2 \sqrt {2} d \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}\right )}{b \sqrt {g \cos (e+f x)}}\right )}{b}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )}{b}-\frac {a d \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} \sqrt {d} \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (b-\sqrt {b^2-a^2}\right )}+\frac {2 \sqrt {2} \sqrt {d} \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (\sqrt {b^2-a^2}+b\right )}\right )}{b \sqrt {g \cos (e+f x)}}\right )}{b}\) |
-((a*d*(-((a*d*Sqrt[Cos[e + f*x]]*((2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*Sqr t[d]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/( Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/((b - Sqrt[-a^2 + b^2])*f) + (2*Sqr t[2]*(1 + b/Sqrt[-a^2 + b^2])*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2] )), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(( b + Sqrt[-a^2 + b^2])*f)))/(b*Sqrt[g*Cos[e + f*x]])) + (2*d^2*g*((-(ArcTan [1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]]) ]/(Sqrt[2]*Sqrt[d]*Sqrt[g])) + ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*g) - (-1/2*Log[d - (Sqrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + d*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[d + (Sqrt[2]*Sqr t[d]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + d*Tan[e + f*x]]/ (2*Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*g)))/(b*f)))/b) + (d*(-((d*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(f*g)) + (d^2*EllipticF[e - Pi/4 + f*x, 2]*Sqr t[Sin[2*e + 2*f*x]])/(2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])))/b
3.15.30.3.1 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[((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] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
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[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[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q)) Subst[Int[1/((d*(b + q) + a*x^2)*Sq rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Simp[2*Sqrt[2]*d*((b - q)/(f*q)) Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.) ]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Cos[e + f *x]]/Sqrt[g*Cos[e + f*x]] Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[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. 3414 vs. \(2 (522 ) = 1044\).
Time = 3.32 (sec) , antiderivative size = 3415, normalized size of antiderivative = 5.54
1/f*csc(f*x+e)*((-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f* x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f *x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a*b*cos(f*x+e)+I*(-a^2+b^ 2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2) *(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2 ),1/2-1/2*I,1/2*2^(1/2))*a^2*cos(f*x+e)-I*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+cs c(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e) )^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2)) *a^2*cos(f*x+e)+I*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc( f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot (f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a*b-I*(-a^2+b^2)^(1/2)* (-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f* x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2 *I,1/2*2^(1/2))*a*b+(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-cs c(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-c ot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a*b*cos(f*x+e)-cos(f* x+e)*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f* x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f* x+e)+1)^(1/2),1/2*2^(1/2))*a*b+cos(f*x+e)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2) *(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*Ellipt...
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {g \cos \left (f x + e\right )} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \]
\[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {g \cos \left (f x + e\right )} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {g\,\cos \left (e+f\,x\right )}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]