Integrand size = 29, antiderivative size = 771 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} (5 b c+3 d) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 b (b c-3 d) f}+\frac {\sqrt {c+d} \left (18 b c d-9 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 b^2 \sqrt {3+b} d f}+\frac {(b c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {3+b \sin (e+f x)}}-\frac {(5 b c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {3+b \sin (e+f x)}}+\frac {(3+b)^{3/2} (5 b c-3 d+2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 b^2 \sqrt {c+d} f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {3+b \sin (e+f x)}} \]
1/4*(6*a*b*c*d-a^2*d^2+b^2*(3*c^2+4*d^2))*EllipticPi((a+b)^(1/2)*(c+d*sin( f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c +d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b* c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e)) /(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^2/d/f/(a+b)^(1/2)+1/4*(c-d)*(a*d+5*b*c)*E llipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^( 1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(a+b)^(1 /2)*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)* ((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b/(-a*d+b*c)/f-1/ 2*b*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+b*sin(f*x+e))^(1/2)+1/4*(a+b)^( 3/2)*(-a*d+5*b*c+2*b*d)*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b) ^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)* (c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)* (-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/b^2/f/(c+d)^(1/2 )+1/2*(-a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+b*sin(f*x+e))^(1/2 )-1/4*(a*d+5*b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+b*sin(f*x+e))^(1/ 2)
Leaf count is larger than twice the leaf count of optimal. \(1855\) vs. \(2(771)=1542\).
Time = 13.62 (sec) , antiderivative size = 1855, normalized size of antiderivative = 2.41 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx =\text {Too large to display} \]
-1/2*(d*Cos[e + f*x]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + ((-4*(-(b*c) + 3*d)*(24*c^2 + 7*b*c*d + 9*d^2)*Sqrt[((c + d)*Cot[(-e + P i/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3 *d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt [((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d) ])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4 *(-(b*c) + 3*d)*(8*b*c^2 + 36*c*d + 4*b*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/ 2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d ))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[( (-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]) /((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sq rt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d )/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin [e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d) )]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e ...
Time = 4.16 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3300, 27, 3042, 3526, 27, 3042, 3540, 25, 3042, 3532, 25, 3042, 3290, 3477, 3042, 3297, 3475}
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.
| \(\displaystyle \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3300 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 b d (b c+a d) \sin ^2(e+f x)+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+3 b d a-b^2 c\right )\right )}{2 (a+b \sin (e+f x))^{3/2}}dx}{2 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 b d (b c+a d) \sin ^2(e+f x)+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+3 b d a-b^2 c\right )\right )}{(a+b \sin (e+f x))^{3/2}}dx}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 b d (b c+a d) \sin (e+f x)^2+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+3 b d a-b^2 c\right )\right )}{(a+b \sin (e+f x))^{3/2}}dx}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {2 \int -\frac {b \left (a^2-b^2\right ) d^2 (5 b c+a d) \sin ^2(e+f x)+2 b \left (a^2-b^2\right ) d \left (3 a c d+b \left (2 c^2+d^2\right )\right ) \sin (e+f x)+b \left (a^2-b^2\right ) d \left (4 a c^2+b d c+a d^2\right )}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (a^2-b^2\right ) d^2 (5 b c+a d) \sin ^2(e+f x)+2 b \left (a^2-b^2\right ) d \left (3 a c d+b \left (2 c^2+d^2\right )\right ) \sin (e+f x)+b \left (a^2-b^2\right ) d \left (b c d+a \left (4 c^2+d^2\right )\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (a^2-b^2\right ) d^2 (5 b c+a d) \sin (e+f x)^2+2 b \left (a^2-b^2\right ) d \left (3 a c d+b \left (2 c^2+d^2\right )\right ) \sin (e+f x)+b \left (a^2-b^2\right ) d \left (b c d+a \left (4 c^2+d^2\right )\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {-b \left (a^2-b^2\right ) \left (\left (3 c^2+4 d^2\right ) b^2+6 a c d b-a^2 d^2\right ) \sin ^2(e+f x) d^2+b \left (a^2-b^2\right ) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )-6 b c d a+5 b^2 c^2\right ) d^2-2 b \left (a^2-b^2\right ) \left (5 c d a^2+3 b \left (c^2+d^2\right ) a+b^2 c d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-b \left (a^2-b^2\right ) \left (\left (3 c^2+4 d^2\right ) b^2+6 a c d b-a^2 d^2\right ) \sin ^2(e+f x) d^2+b \left (a^2-b^2\right ) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )-6 b c d a+5 b^2 c^2\right ) d^2-2 b \left (a^2-b^2\right ) \left (5 c d a^2+3 b \left (c^2+d^2\right ) a+b^2 c d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-b \left (a^2-b^2\right ) \left (\left (3 c^2+4 d^2\right ) b^2+6 a c d b-a^2 d^2\right ) \sin (e+f x)^2 d^2+b \left (a^2-b^2\right ) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )-6 b c d a+5 b^2 c^2\right ) d^2-2 b \left (a^2-b^2\right ) \left (5 c d a^2+3 b \left (c^2+d^2\right ) a+b^2 c d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int -\frac {b \left (a^2-b^2\right )^2 d^2 (b c-a d) (5 b c-a d)-2 b^2 \left (a^2-b^2\right )^2 d^3 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {d^2 \left (a^2-b^2\right ) \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {d^2 \left (a^2-b^2\right ) \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {\int \frac {b \left (a^2-b^2\right )^2 d^2 (b c-a d) (5 b c-a d)-2 b^2 \left (a^2-b^2\right )^2 d^3 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {d^2 \left (a^2-b^2\right ) \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {\int \frac {b \left (a^2-b^2\right )^2 d^2 (b c-a d) (5 b c-a d)-2 b^2 \left (a^2-b^2\right )^2 d^3 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {b \left (a^2-b^2\right )^2 d^2 (b c-a d) (5 b c-a d)-2 b^2 \left (a^2-b^2\right )^2 d^3 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {b d^2 (a-b) (a+b)^2 (b c-a d) (-a d+5 b c+2 b d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^2 d^2 (a-b) (a+b)^2 (b c-a d) (a d+5 b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {b d^2 (a-b) (a+b)^2 (b c-a d) (-a d+5 b c+2 b d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^2 d^2 (a-b) (a+b)^2 (b c-a d) (a d+5 b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {2 b d^2 (a-b) (a+b)^{5/2} (-a d+5 b c+2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}-b^2 d^2 (a-b) (a+b)^2 (b c-a d) (a d+5 b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b d \left (a^2-b^2\right ) (a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}+\frac {2 d (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {2 d (b c-a d) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}+\frac {-\frac {b \left (a^2-b^2\right ) d (5 b c+a d) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {2 \left (a^2-b^2\right ) d \sqrt {c+d} \left (\left (3 c^2+4 d^2\right ) b^2+6 a c d b-a^2 d^2\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b \sqrt {a+b} f}-\frac {\frac {2 (a-b) b d^2 (5 b c-a d+2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{5/2}}{\sqrt {c+d} f}+\frac {2 (a-b) b^2 (c-d) d^2 \sqrt {c+d} (5 b c+a d) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) (a+b)^{3/2}}{(b c-a d) f}}{b^2}}{2 d}}{b \left (a^2-b^2\right )}}{4 d}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\) |
-1/2*(b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*Sqrt[a + b*Sin[e + f*x ]]) + ((2*d*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) + (-((b*(a^2 - b^2)*d*(5*b*c + a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]])) - ((-2*(a^2 - b^2)*d*Sqrt [c + d]*(6*a*b*c*d - a^2*d^2 + b^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d)) /((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*S qrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f* x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))) ]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*Sqrt[a + b]*f) - ((2*(a - b)*b^2*(a + b)^(3/2)*(c - d)*d^2*Sqrt[c + d]*(5*b*c + a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d *Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d)) /((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/( (c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((b*c - a*d)*f) + (2*(a - b)*b*(a + b)^(5/2)*d^2*(5*b*c - a*d + 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[c ...
3.8.66.3.1 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) I nt[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*( m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Result contains complex when optimal does not.
Time = 11.78 (sec) , antiderivative size = 247614, normalized size of antiderivative = 321.16
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]