Integrand size = 37, antiderivative size = 577 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}-\frac {g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {g^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b \sqrt {d} f}+\frac {g^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b \sqrt {d} f}+\frac {g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
1/2*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))/b/f*2^(1/2)/d^(1/2)-1/2*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))/b/f*2^(1/2)/d^(1/2)-1/4*g^ (3/2)*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/f*2^(1/2)/d^(1/2)+1/4*g^(3/2)*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/f*2^ (1/2)/d^(1/2)+2*g^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)*(-a^2+b^2)^(1/2)*cos(f*x+e)^(1/2 )/a/b/f/d^(1/2)/(g*cos(f*x+e))^(1/2)-2*g^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)*(-a^2+b^2 )^(1/2)*cos(f*x+e)^(1/2)/a/b/f/d^(1/2)/(g*cos(f*x+e))^(1/2)-g^2*(sin(e+1/4 *Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*s in(2*f*x+2*e)^(1/2)/a/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 = 12.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.31 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \left (b \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-a \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) (g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)} \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{5 \left (a^2-b^2\right ) d f g \sqrt [4]{\sin ^2(e+f x)} (a+b \sin (e+f x))} \]
(2*(b*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^ 2 + b^2)] - a*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x] ^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sqrt [Sin[e + f*x]^2]))/(5*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(1/4)*(a + b*Sin[ e + f*x]))
Time = 2.40 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.05, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.568, Rules used = {3042, 3379, 3042, 3317, 3042, 3053, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3120, 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.
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3379 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \int \frac {b-a \sin (e+f x)}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{a b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \int \frac {b-a \sin (e+f x)}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{a b}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \int \frac {d \tan (e+f x)}{g \left (\tan ^2(e+f x) d^2+d^2\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}+\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} g}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {d} g}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 3387 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 3386 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} d \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}+\frac {2 \sqrt {2} d \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}\right )}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} \sqrt {d} \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (b-\sqrt {b^2-a^2}\right )}+\frac {2 \sqrt {2} \sqrt {d} \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (\sqrt {b^2-a^2}+b\right )}\right )}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\) |
((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*((2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*S qrt[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*S qrt[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)))/(a*b*d*Sqrt[g*Cos[e + f*x]]) + (g^2*((-2*a*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]])/Sqr t[g*Cos[e + f*x]] + d*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[d + (S qrt[2]*Sqrt[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)))/f + (b*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f* x]])))/(a*b)
3.15.18.3.1 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[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/(a*b) Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(b - a*Sin[e + f*x]), x], x ] + Simp[g^2*((a^2 - b^2)/(a*b*d)) Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e + f*x])^(n + 1)/(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] && (LtQ[n, -1] || (EqQ[p, 3/2] && EqQ[n, -2^(-1)]))
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (490 ) = 980\).
Time = 1.82 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.04
(-1/2+1/2*I)/f*(a-b)*(-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b +(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2+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^2+b^2)^(1/2)*a+I*EllipticPi((-cot(f *x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2+I*Ell ipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^( 1/2))*b^2-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^( 1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b-I*EllipticPi((-cot(f*x+e)+csc(f*x+ e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a^2-I*EllipticPi((-cot( f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^ 2)^(1/2)*b+2*I*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*(-a ^2+b^2)^(1/2)*b-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+ b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-I*EllipticPi((-cot(f*x+e)+cs c(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)* a+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+b^2)^(1/2)*a-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^ 2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-EllipticPi((-cot(f*x+e)+csc(f* x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b-El lipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^ (1/2))*a^2+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1 /2)+a),1/2*2^(1/2))*b^2-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/...
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {\left (g \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {d \sin {\left (e + f x \right )}} \left (a + b \sin {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {d\,\sin \left (e+f\,x\right )}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]