Integrand size = 37, antiderivative size = 380 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \sqrt {2} b^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{(-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^2 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{(-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 a \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)}}-\frac {2 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {2 b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d f g^2 \sqrt {\sin (2 e+2 f x)}} \]
-2*b*(d*sin(f*x+e))^(3/2)/(a^2-b^2)/d^2/f/g/(g*cos(f*x+e))^(1/2)+2*b^2*Ell ipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a +b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/(-a+b)^(3/2)/(a+b)^(3/2)/f/g^(3/2)/( d*sin(f*x+e))^(1/2)-2*b^2*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f *x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/(-a+b)^( 3/2)/(a+b)^(3/2)/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+2*a*(d*sin(f*x+e))^(1/2)/( a^2-b^2)/d/f/g/(g*cos(f*x+e))^(1/2)-2*b*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+ 1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*s in(f*x+e))^(1/2)/(a^2-b^2)/d/f/g^2/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 22.13 (sec) , antiderivative size = 1279, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \cos (e+f x) \sin (e+f x) (a-b \sin (e+f x))}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}+\frac {b \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} \left (\frac {4 a \left (-b \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}+\frac {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (56 b \left (-3 a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+24 b \left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+21 a^{3/2} \left (4 \sqrt {2} a^{3/2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+2 \sqrt {2} a^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2 \sqrt {2} a^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {2 \sqrt {2} a^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} a^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {8 \sqrt {a} b \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {1+\tan ^2(e+f x)}}\right )\right )}{84 a^2 b \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{(-a+b) (a+b) f (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \]
(2*Cos[e + f*x]*Sin[e + f*x]*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]) + (b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2 *Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a ^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*Arc Tan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2] *Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4 )*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[ 1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^( 1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f *x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^ 2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2] *a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^ (3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a...
Time = 2.16 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.92, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 3383, 3042, 3317, 3042, 3043, 3051, 3042, 3052, 3042, 3119, 3385, 3042, 3384, 993, 1542}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3383 |
| \(\displaystyle \frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3051 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {4 \sqrt {2} b^2 \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}+\frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {4 \sqrt {2} b^2 \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}+\frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {4 \sqrt {2} b^2 \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}+\frac {\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}-\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}}{a^2-b^2}\) |
(4*Sqrt[2]*b^2*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g *Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(2*Sqrt[-a + b]*Sqrt[a + b] *Sqrt[g]))*Sqrt[Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[d*Sin[e + f*x]]) + (( 2*a*Sqrt[d*Sin[e + f*x]])/(d*f*g*Sqrt[g*Cos[e + f*x]]) - (b*((2*(d*Sin[e + f*x])^(3/2))/(d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*Ellip ticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(f*g^2*Sqrt[Sin[2*e + 2*f*x] ])))/d)/(a^2 - b^2)
3.15.39.3.1 Defintions of rubi rules used
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[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*Sin[e + f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/ (a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] & & NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) /(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Sin[e + f*x ])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m , -1] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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[1/(a^2 - b^2 ) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] - Simp[b^2/(g^2*(a^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] && LtQ[p, -1]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2 , 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1203\) vs. \(2(347)=694\).
Time = 2.31 (sec) , antiderivative size = 1204, normalized size of antiderivative = 3.17
1/f/(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(1/2)*(2 *EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b*(-a^2+b^2)^(1 /2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*( -csc(f*x+e)+cot(f*x+e))^(1/2)+2*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2) ,1/2*2^(1/2))*b^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*c ot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-4*EllipticE(( -cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b*(-a^2+b^2)^(1/2)*(-cot(f* x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+ cot(f*x+e))^(1/2)-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*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1) ^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)- 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*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f* x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+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))*a*b^2*(-c ot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f* x+e)+cot(f*x+e))^(1/2)+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^3*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2* cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-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))*...
Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {1}{\sqrt {d \sin {\left (e + f x \right )}} \left (g \cos {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \sin {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
\[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {1}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]