Integrand size = 29, antiderivative size = 861 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (42 b c d-27 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{24 b^2 (b c-3 d) f}-\frac {\sqrt {c+d} \left (45 b c d^2-27 d^3-3 b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{8 b^3 \sqrt {3+b} d f}-\frac {\left (42 b c d-27 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {3+b \sin (e+f x)}}-\frac {(13 b c-9 d) d \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(3+b)^{3/2} \left (27 d^2-18 b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f} \]
-1/8*(5*a^2*b*c*d^2-a^3*d^3-a*b^2*d*(15*c^2+4*d^2)-5*b^3*(c^3+4*c*d^2))*El lipticPi((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^3/d/f/(a+b) ^(1/2)+1/24*(c-d)*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16*d^2))*EllipticE((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^2/(-a*d+b*c)/f+1/24*(a+b)^( 3/2)*(3*a^2*d^2-6*a*b*d*(2*c+d)+b^2*(33*c^2+26*c*d+16*d^2))*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^3/f/(c+d)^(1/2)-1/3*d^2*cos(f*x+e)*(a+b*sin(f*x+e))^ (3/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/24*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16 *d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/f/(a+b*sin(f*x+e))^(1/2)-1/12*d *(-3*a*d+13*b*c)*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)/ b/f
Leaf count is larger than twice the leaf count of optimal. \(1945\) vs. \(2(861)=1722\).
Time = 8.69 (sec) , antiderivative size = 1945, normalized size of antiderivative = 2.26 \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \]
((-4*(-(b*c) + 3*d)*(144*b*c^3 + 59*b^2*c^2*d + 174*b*c*d^2 - 9*d^3 + 16*b ^2*d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcS in[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*Si n[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)*(48*b^2*c^3 + 276*b*c^2*d - 36*c*d^2 + 76*b^2*c*d^2 + 84*b*d^3)*((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]]) - (Sqrt[((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 + Pi/2...
Time = 4.33 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.05, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 3272, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 25, 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.
| \(\displaystyle \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (a d^3+(13 b c-3 a d) \sin ^2(e+f x) d^2+2 \left (9 b c^2-a d c+2 b d^2\right ) \sin (e+f x) d+3 b c \left (2 c^2+d^2\right )\right )}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (a d^3+(13 b c-3 a d) \sin ^2(e+f x) d^2+2 \left (9 b c^2-a d c+2 b d^2\right ) \sin (e+f x) d+3 b c \left (2 c^2+d^2\right )\right )}{\sqrt {c+d \sin (e+f x)}}dx}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (a d^3+(13 b c-3 a d) \sin (e+f x)^2 d^2+2 \left (9 b c^2-a d c+2 b d^2\right ) \sin (e+f x) d+3 b c \left (2 c^2+d^2\right )\right )}{\sqrt {c+d \sin (e+f x)}}dx}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int \frac {d^2 \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (-\left (\left (12 c^3+19 d^2 c\right ) b^2\right )-a d \left (23 c^2+7 d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d+a b \left (24 c^3+22 d^2 c\right )\right )}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {d^2 \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (-\left (\left (12 c^3+19 d^2 c\right ) b^2\right )-a d \left (23 c^2+7 d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d+a b \left (24 c^3+22 d^2 c\right )\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {d^2 \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2-2 d \left (-\left (\left (12 c^3+19 d^2 c\right ) b^2\right )-a d \left (23 c^2+7 d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)+d \left (a^2 d^3+13 b^2 c^2 d+a b \left (24 c^3+22 d^2 c\right )\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {3 \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \sin ^2(e+f x) d^2+\left (c \left (33 c^2+16 d^2\right ) b^3-a d \left (45 c^2+16 d^2\right ) b^2-a^2 c \left (48 c^2+61 d^2\right ) b+a^3 d^3\right ) d^2-2 \left (c d^2 a^3+b d \left (32 c^2+15 d^2\right ) a^2+b^2 c \left (15 c^2+44 d^2\right ) a+13 b^3 c^2 d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {3 \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \sin ^2(e+f x) d^2+\left (c \left (33 c^2+16 d^2\right ) b^3-a d \left (45 c^2+16 d^2\right ) b^2-a^2 c \left (48 c^2+61 d^2\right ) b+a^3 d^3\right ) d^2-2 \left (c d^2 a^3+b d \left (32 c^2+15 d^2\right ) a^2+b^2 c \left (15 c^2+44 d^2\right ) a+13 b^3 c^2 d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {3 \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \sin (e+f x)^2 d^2+\left (c \left (33 c^2+16 d^2\right ) b^3-a d \left (45 c^2+16 d^2\right ) b^2-a^2 c \left (48 c^2+61 d^2\right ) b+a^3 d^3\right ) d^2-2 \left (c d^2 a^3+b d \left (32 c^2+15 d^2\right ) a^2+b^2 c \left (15 c^2+44 d^2\right ) a+13 b^3 c^2 d\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int -\frac {\left (a^2-b^2\right ) d^2 (b c-a d) \left (33 b^2 c^2-12 a b d c+3 a^2 d^2+16 b^2 d^2\right )-2 b \left (a^2-b^2\right ) d^3 (13 b c-3 a d) (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3 d^2 \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {\int -\frac {2 b \left (a^2-b^2\right ) (13 b c-3 a d) (b c-a d) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (33 c^2+16 d^2\right ) b^2\right )+12 a c d b-3 a^2 d^2\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (13 b c-3 a d) (b c-a d) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (33 c^2+16 d^2\right ) b^2\right )+12 a c d b-3 a^2 d^2\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (13 b c-3 a d) (b c-a d) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (33 c^2+16 d^2\right ) b^2\right )+12 a c d b-3 a^2 d^2\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (13 b c-3 a d) (b c-a d) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (33 c^2+16 d^2\right ) b^2\right )+12 a c d b-3 a^2 d^2\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 d \sqrt {c+d} \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b d^2 (a+b) (b c-a d) \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d^2 (a+b) (b c-a d) \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 d \sqrt {c+d} \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b d^2 (a+b) (b c-a d) \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d^2 (a+b) (b c-a d) \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 d \sqrt {c+d} \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b d^2 (a+b) (b c-a d) \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {2 d^2 (a+b)^{3/2} \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}}{b^2}+\frac {6 d \sqrt {c+d} \left (-a^3 d^3+5 a^2 b c d^2-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {d \left (-3 a^2 d^2+14 a b c d+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {-\frac {d \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {6 d \sqrt {c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {-\frac {2 b \sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) d^2}{(b c-a d) f}-\frac {2 (a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) d^2}{\sqrt {c+d} f}}{b^2}}{2 d}}{4 d}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}\) |
-1/3*(d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]] )/(b*f) + (-1/2*(d*(13*b*c - 3*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])/f + (-((d*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f* x]])) - ((6*d*Sqrt[c + d]*(5*a^2*b*c*d^2 - a^3*d^3 - a*b^2*d*(15*c^2 + 4*d ^2) - 5*b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(S qrt[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^ 2*Sqrt[a + b]*f) + ((-2*b*Sqrt[a + b]*(c - d)*d^2*Sqrt[c + d]*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*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)^(3/2)*d^2*(3*a^2*d^2 - 6*a*b*d*(2*c + d) + b^2*(33*c^2 + 26*c*d + 16*d^2))*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...
3.8.65.3.1 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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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_)])/(((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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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 = 18.53 (sec) , antiderivative size = 364669, normalized size of antiderivative = 423.54
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx=\int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]