Integrand size = 37, antiderivative size = 814 \[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\frac {3 d^{3/2} g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^3 f}-\frac {3 d^{3/2} g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{4 \sqrt {2} b f}-\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b^3 f}+\frac {3 d^{3/2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} \left (\sqrt {d}+\sqrt {d} \tan (e+f x)\right )}\right )}{4 \sqrt {2} b f}+\frac {\left (a^2-b^2\right ) d^{3/2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} \left (\sqrt {d}+\sqrt {d} \tan (e+f x)\right )}\right )}{\sqrt {2} b^3 f}+\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a \sqrt {-a^2+b^2} d^{3/2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b^3 f \sqrt {g \cos (e+f x)}}-\frac {a d g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{b^2 f}+\frac {g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 b f}+\frac {a d^2 g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \] Output:
3/8*d^(3/2)*g^(3/2)*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))*2^(1/2)/b/f+1/2*(a^2-b^2)*d^(3/2)*g^(3/2)*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))*2^(1/2)/ b^3/f-3/8*d^(3/2)*g^(3/2)*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))*2^(1/2)/b/f-1/2*(a^2-b^2)*d^(3/2)*g^(3/2)*arct an(1+2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f*x+e))^(1/2))*2^ (1/2)/b^3/f+3/8*d^(3/2)*g^(3/2)*arctanh(2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/ 2)/(g*cos(f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*tan(f*x+e)))*2^(1/2)/b/f+1/2*(a^2 -b^2)*d^(3/2)*g^(3/2)*arctanh(2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos( f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*tan(f*x+e)))*2^(1/2)/b^3/f+2*2^(1/2)*a*(-a^ 2+b^2)^(1/2)*d^(3/2)*g^2*cos(f*x+e)^(1/2)*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)/b^3/f/(g*cos(f*x+e ))^(1/2)-2*2^(1/2)*a*(-a^2+b^2)^(1/2)*d^(3/2)*g^2*cos(f*x+e)^(1/2)*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)/b^3/f/(g*cos(f*x+e))^(1/2)-a*d*g*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e) )^(1/2)/b^2/f+1/2*g*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(3/2)/b/f+1/2*a*d^ 2*g^2*InverseJacobiAM(e-1/4*Pi+f*x,2^(1/2))*sin(2*f*x+2*e)^(1/2)/b^2/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 = 51.99 (sec) , antiderivative size = 1898, normalized size of antiderivative = 2.33 \[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx =\text {Too large to display} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f *x]),x]
Output:
((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(d*Sin[e + f*x])^(3/2))/(2*b*f) - ((g *Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*((10*b*(a^2 - b^2)*Sqrt[Cos[e + f*x]]*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((b*AppellF1[1/4, -3/4, 1, 5/4, C os[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2] )/(-5*(a^2 - b^2)*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (4*b^2*AppellF1[5/4, -3/4, 2, 9/4, Cos[e + f*x]^2 , (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2) + (a*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^ 2 + b^2)])/(5*(a^2 - b^2)*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2 *Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, -1/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2))*Sin[e + f*x]^(5/2))/((1 - Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))*(a + b*Sin[e + f*x])) + (2*a*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*A rcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcT an[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 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]] - 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*Ap...
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.
| \(\displaystyle \int \frac {(d \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}}{a+b \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3380 |
| \(\displaystyle \frac {g^2 \int \frac {(d \sin (e+f x))^{3/2} (a-b \sin (e+f x))}{\sqrt {g \cos (e+f x)}}dx}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \int \frac {(d \sin (e+f x))^{3/2} (a-b \sin (e+f x))}{\sqrt {g \cos (e+f x)}}dx}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {g^2 \left (a \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}}dx-\frac {b \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)}}dx-\frac {b \int \frac {(d \sin (e+f x))^{5/2}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3}{4} d^2 \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3}{4} d^2 \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3}{4} d^2 \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3}{4} d^2 \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \int \frac {d \tan (e+f x)}{g \left (\tan ^2(e+f x) d^2+d^2\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}+\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} g}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {d} g}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b^2}\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \int \frac {d \tan (e+f x)}{g \left (\tan ^2(e+f x) d^2+d^2\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \left (\frac {\int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}+\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {\sin (2 e+2 f x)}}{2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {d \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 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 (\tan (e+f x) d+d+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\tan (e+f x) d+d-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 f g}\right )}{d}\right )}{b^2}-\frac {\left (a^2-b^2\right ) g^2 \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^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {\sin (2 e+2 f x)}}{2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {d \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 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 (\tan (e+f x) d+d+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\tan (e+f x) d+d-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 f g}\right )}{d}\right )}{b^2}-\frac {\left (a^2-b^2\right ) g^2 \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^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {\sin (2 e+2 f x)}}{2 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {d \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 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 (\tan (e+f x) d+d+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (\tan (e+f x) d+d-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g \cos (e+f x)}}\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{2 f g}\right )}{d}\right )}{b^2}-\frac {\left (a^2-b^2\right ) g^2 \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^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {g^2 \left (a \left (\frac {d^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {d \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g}\right )-\frac {b \left (\frac {3 d^3 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{2 f}-\frac {d (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}{2 f g}\right )}{d}\right )}{b^2}-\frac {g^2 \left (a^2-b^2\right ) \left (\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{b f}-\frac {a d \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{b}\right )}{b^2}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]),x ]
Output:
$Aborted
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[(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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/b^2 In t[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] - Simp[g^2*((a^2 - b^2)/b^2) Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e + f*x ])^n/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a ^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1]
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. 2066 vs. \(2 (656 ) = 1312\).
Time = 4.30 (sec) , antiderivative size = 2067, normalized size of antiderivative = 2.54
Input:
int((g*cos(f*x+e))^(3/2)*(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x,method=_R ETURNVERBOSE)
Output:
1/4/f*sec(f*x+e)*csc(f*x+e)*(d*sin(f*x+e))^(1/2)*(g*cos(f*x+e))^(1/2)*(I*( -4*cos(f*x+e)-4)*(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc( f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((cs c(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a^2+I*(-1-cos(f*x+e))* (-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1/2* 2^(1/2))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1 /2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*b^2+I*(4*cos(f*x+e)+4)*(-a^2+b^2)^(1/2) *EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(csc(f* x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e) +cot(f*x+e))^(1/2)*a^2+I*(1+cos(f*x+e))*(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f *x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e ))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2)) *b^2+(-4*cos(f*x+e)-4)*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+ 1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f* x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*a^2+(1+cos(f*x+e ))*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1 /2*2^(1/2))*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2) ^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*b^2+(4*cos(f*x+e)+4)*(-a^2+b^2)^(1/2 )*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-c sc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),...
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(d*sin(f*x+e))**(3/2)/(a+b*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*(d*sin(f*x + e))^(3/2)/(b*sin(f*x + e) + a), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al gorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*(d*sin(f*x + e))^(3/2)/(b*sin(f*x + e) + a), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{a+b\,\sin \left (e+f\,x\right )} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(3/2))/(a + b*sin(e + f*x)),x )
Output:
int(((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(3/2))/(a + b*sin(e + f*x)), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx=\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )}{\sin \left (f x +e \right ) b +a}d x \right ) d g \] Input:
int((g*cos(f*x+e))^(3/2)*(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x)
Output:
sqrt(g)*sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x))*cos(e + f*x)*si n(e + f*x))/(sin(e + f*x)*b + a),x)*d*g