Integrand size = 39, antiderivative size = 840 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\frac {(c-d) \sqrt {c+d} \left (2 A b^2 c-2 a b B c-2 a A b d+3 a^2 B d-b^2 B d\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt {a+b} (b c-a d) f}+\frac {\sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt {a+b} f}+\frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\left (2 A b (b c-a d)-B \left (2 a b c-3 a^2 d+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} \left (2 A b (b (c-2 d)+a d)-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt {c+d} f} \] Output:
(c-d)*(c+d)^(1/2)*(-2*A*a*b*d+2*A*b^2*c+3*B*a^2*d-2*B*a*b*c-B*b^2*d)*Ellip ticE((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*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)*(a+b*sin(f*x+e))/(a-b)/b^2/(a+b)^(1/2)/(-a*d+b*c)/f+(c+d)^(1/2 )*(2*A*b*d-3*B*a*d+3*B*b*c)*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*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)*(a+b*sin(f*x +e))/b^3/(a+b)^(1/2)/f+2*(A*b-B*a)*(-a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^ (1/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^(1/2)-(2*A*b*(-a*d+b*c)-B*(-3*a^2*d+2 *a*b*c+b^2*d))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/(a+b*sin(f* x+e))^(1/2)+(a+b)^(1/2)*(2*A*b*(b*(c-2*d)+a*d)-B*(3*a^2*d-6*a*b*d+b^2*(2*c +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)*((-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)*(c+d*sin(f*x+e))/(a-b)/b^3/(c+d)^(1/2)/f
Leaf count is larger than twice the leaf count of optimal. \(2042\) vs. \(2(840)=1680\).
Time = 9.39 (sec) , antiderivative size = 2042, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x])^(3/2),x]
Output:
(-2*(A*b^2*c*Cos[e + f*x] - a*b*B*c*Cos[e + f*x] - a*A*b*d*Cos[e + f*x] + a^2*B*d*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(b*(-a^2 + b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((-4*(-(b*c) + a*d)*(2*a*A*b*c^2 - 2*b^2*B*c^2 - 2*A*b ^2*c*d + 2*a*b*B*c*d + a^2*B*d^2 - b^2*B*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f *x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d) )/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[(( -a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/ ((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(- (b*c) + a*d)*(2*A*b^2*c^2 - 2*a*b*B*c^2 + 4*a^2*B*c*d - 4*b^2*B*c*d - 2*A* b^2*d^2 + 2*a*b*B*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d )]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/ 2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt [a + 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) + a*d)/((a + b)*d), ArcSin[ Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c)...
Time = 4.46 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3468, 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.
| \(\displaystyle \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {2 \int \frac {c (B c+2 A d) b^2-a \left (2 B c d+A \left (c^2+d^2\right )\right ) b+a^2 B d^2-d \left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) \sin ^2(e+f x)-\left (A \left (c^2-d^2\right ) b^2+B \left (2 c d a^2-b \left (c^2-d^2\right ) a-2 b^2 c d\right )\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {c (B c+2 A d) b^2-a \left (2 B c d+A \left (c^2+d^2\right )\right ) b+a^2 B d^2-d \left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) \sin ^2(e+f x)-\left (A \left (c^2-d^2\right ) b^2+B \left (2 c d a^2-b \left (c^2-d^2\right ) a-2 b^2 c d\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {c (B c+2 A d) b^2-a \left (2 B c d+A \left (c^2+d^2\right )\right ) b+a^2 B d^2-d \left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) \sin (e+f x)^2-\left (A \left (c^2-d^2\right ) b^2+B \left (2 c d a^2-b \left (c^2-d^2\right ) a-2 b^2 c d\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\int -\frac {\left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d) \sin ^2(e+f x)+2 \left (a^2-b^2\right ) c d (b B c+2 A b d-a B d) \sin (e+f x)+\left (a^2-b^2\right ) d \left (2 A b c^2-B d (b c-a d)\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}+\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {\left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d) \sin ^2(e+f x)+2 \left (a^2-b^2\right ) c d (b B c+2 A b d-a B d) \sin (e+f x)+\left (a^2-b^2\right ) d \left (2 A b c^2-B d (b c-a d)\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {\left (a^2-b^2\right ) d^2 (3 b B c+2 A b d-3 a B d) \sin (e+f x)^2+2 \left (a^2-b^2\right ) c d (b B c+2 A b d-a B d) \sin (e+f x)+\left (a^2-b^2\right ) d \left (2 A b c^2-B d (b c-a d)\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {d^2 \left (a^2-b^2\right ) (-3 a B d+2 A b d+3 b B c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {\int -\frac {\left (a^2-b^2\right ) d (b c-a d) \left (\left (3 a^2+b^2\right ) B d-2 A b (b c+a d)\right )-2 b \left (a^2-b^2\right ) d (b c-a d) (b B c+2 A b d-3 a B d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {d^2 \left (a^2-b^2\right ) (-3 a B d+2 A b d+3 b B c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {\int \frac {\left (a^2-b^2\right ) d (b c-a d) \left (\left (3 a^2+b^2\right ) B d-2 A b (b c+a d)\right )-2 b \left (a^2-b^2\right ) d (b c-a d) (b B c+2 A b d-3 a B d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {d^2 \left (a^2-b^2\right ) (-3 a B d+2 A b d+3 b B c) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {\int \frac {\left (a^2-b^2\right ) d (b c-a d) \left (\left (3 a^2+b^2\right ) B d-2 A b (b c+a d)\right )-2 b \left (a^2-b^2\right ) d (b c-a d) (b B c+2 A b d-3 a B d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \sec (e+f x) (-3 a B d+2 A b d+3 b B c) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}-\frac {\int \frac {\left (a^2-b^2\right ) d (b c-a d) \left (\left (3 a^2+b^2\right ) B d-2 A b (b c+a d)\right )-2 b \left (a^2-b^2\right ) d (b c-a d) (b B c+2 A b d-3 a B d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \sec (e+f x) (-3 a B d+2 A b d+3 b B c) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}-\frac {b d (a+b) (b c-a d) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d (a+b) (b c-a d) \left (2 A b (a d+b (c-2 d))-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \sec (e+f x) (-3 a B d+2 A b d+3 b B c) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}-\frac {b d (a+b) (b c-a d) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d (a+b) (b c-a d) \left (2 A b (a d+b (c-2 d))-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\cos (e+f x) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 d \left (a^2-b^2\right ) \sqrt {c+d} \sec (e+f x) (-3 a B d+2 A b d+3 b B c) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}-\frac {b d (a+b) (b c-a d) \left (2 A b (b c-a d)-B \left (-3 a^2 d+2 a b c+b^2 d\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {2 d (a+b)^{3/2} \sec (e+f x) \left (2 A b (a d+b (c-2 d))-B \left (3 a^2 d-6 a b d+b^2 (2 c+d)\right )\right ) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 (A b-a B) (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 \left (a^2-b^2\right ) d \sqrt {c+d} (3 b B c+2 A b d-3 a B d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}-\frac {-\frac {2 d \left (2 A b (b (c-2 d)+a d)-B \left (3 d a^2-6 b d a+b^2 (2 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{\sqrt {c+d} f}-\frac {2 b (c-d) d \sqrt {c+d} \left (2 A b (b c-a d)-B \left (-3 d a^2+2 b c a+b^2 d\right )\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt {a+b}}{(b c-a d) f}}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
Input:
Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]) ^(3/2),x]
Output:
(2*(A*b - a*B)*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) - (((2*A*b*(b*c - a*d) - B*(2*a*b*c - 3 *a^2*d + b^2*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]*(3*b*B*c + 2*A*b*d - 3*a*B*d)*E llipticPi[(b*(c + d))/((a + b)*d), 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^2*Sqrt[a + b]*f) - ((-2*b*Sqrt [a + b]*(c - d)*d*Sqrt[c + d]*(2*A*b*(b*c - a*d) - B*(2*a*b*c - 3*a^2*d + b^2*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)^(3/2)*d*(2*A*b*(b*(c - 2*d) + a*d) - B*(3*a^2*d - 6*a*b*d + b^2*(2*c + d)))*EllipticF[ArcSin[(S qrt[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]))/...
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_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((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 - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, 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] && GtQ[m, 1] && LtQ[n, -1]
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_)] + (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 has leaf size over 500,000. Avoiding possible recursion issues.
Time = 78.86 (sec) , antiderivative size = 1089305, normalized size of antiderivative = 1296.79
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1089305\) |
| parts | \(\text {Expression too large to display}\) | \(1235992\) |
Input:
int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x,metho d=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x , algorithm="fricas")
Output:
Timed out
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**(3/2)/(a+b*sin(f*x+e))**(3/2) ,x)
Output:
Integral((A + B*sin(e + f*x))*(c + d*sin(e + f*x))**(3/2)/(a + b*sin(e + f *x))**(3/2), x)
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x , algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(d*sin(f*x + e) + c)^(3/2)/(b*sin(f*x + e) + a)^(3/2), x)
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x , algorithm="giac")
Output:
integrate((B*sin(f*x + e) + A)*(d*sin(f*x + e) + c)^(3/2)/(b*sin(f*x + e) + a)^(3/2), x)
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(a + b*sin(e + f*x)) ^(3/2),x)
Output:
int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(a + b*sin(e + f*x)) ^(3/2), x)
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right ) b +a}d x \right ) d +\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}}{\sin \left (f x +e \right ) b +a}d x \right ) c \] Input:
int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x)
Output:
int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a)*sin(e + f*x))/(sin( e + f*x)*b + a),x)*d + int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a))/(sin(e + f*x)*b + a),x)*c