Integrand size = 39, antiderivative size = 858 \[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 b (A b-a B) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {2 d \left (A \left (a^2 d^2+b^2 \left (3 c^2-4 d^2\right )\right )-B \left (a^2 c d-b^2 c d+3 a b \left (c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (B \left (2 a^2 b c d \left (3 c^2-d^2\right )-2 b^3 c d \left (3 c^2-d^2\right )-a^3 d^2 \left (c^2+3 d^2\right )+a b^2 \left (3 c^4-5 c^2 d^2+6 d^4\right )\right )+A \left (4 a^3 c d^3-4 a b^2 c d^3-a^2 b d^2 \left (9 c^2-5 d^2\right )-b^3 \left (3 c^4-15 c^2 d^2+8 d^4\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^4 f}-\frac {2 \left (B \left (a^2 d^2 (c+3 d)-b^2 c \left (3 c^2+3 c d-2 d^2\right )-6 a b d \left (c^2-d^2\right )\right )-A \left (a^2 d^2 (3 c+d)-6 a b d \left (c^2-d^2\right )+b^2 \left (3 c^3-9 c^2 d-6 c d^2+8 d^3\right )\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^3 f} \] Output:
2*b*(A*b-B*a)*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^(1/2)/(c+ d*sin(f*x+e))^(3/2)+2/3*d*(A*(a^2*d^2+b^2*(3*c^2-4*d^2))-B*(a^2*c*d-b^2*c* d+3*a*b*(c^2-d^2)))*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/(a^2-b^2)/(-a*d+b*c) ^2/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)+2/3*(B*(2*a^2*b*c*d*(3*c^2-d^2)-2*b^ 3*c*d*(3*c^2-d^2)-a^3*d^2*(c^2+3*d^2)+a*b^2*(3*c^4-5*c^2*d^2+6*d^4))+A*(4* a^3*c*d^3-4*a*b^2*c*d^3-a^2*b*d^2*(9*c^2-5*d^2)-b^3*(3*c^4-15*c^2*d^2+8*d^ 4)))*EllipticE((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)^(1/2)/(c-d)^2/(c+d)^(3/2)/(-a *d+b*c)^4/f-2/3*(B*(a^2*d^2*(c+3*d)-b^2*c*(3*c^2+3*c*d-2*d^2)-6*a*b*d*(c^2 -d^2))-A*(a^2*d^2*(3*c+d)-6*a*b*d*(c^2-d^2)+b^2*(3*c^3-9*c^2*d-6*c*d^2+8*d ^3)))*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)^(1/2)/(c-d)^2/(c+d)^(3/2)/(- a*d+b*c)^3/f
Leaf count is larger than twice the leaf count of optimal. \(2837\) vs. \(2(858)=1716\).
Time = 9.30 (sec) , antiderivative size = 2837, normalized size of antiderivative = 3.31 \[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2)),x]
Output:
(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(A*b^4*Cos[e + f*x ] - a*b^3*B*Cos[e + f*x]))/((a^2 - b^2)*(-(b*c) + a*d)^3*(a + b*Sin[e + f* x])) + (2*(-(B*c*d^2*Cos[e + f*x]) + A*d^3*Cos[e + f*x]))/(3*(b*c - a*d)^2 *(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (2*(6*b*B*c^3*d^2*Cos[e + f*x] - 9* A*b*c^2*d^3*Cos[e + f*x] - a*B*c^2*d^3*Cos[e + f*x] + 4*a*A*c*d^4*Cos[e + f*x] - 2*b*B*c*d^4*Cos[e + f*x] + 5*A*b*d^5*Cos[e + f*x] - 3*a*B*d^5*Cos[e + f*x]))/(3*(b*c - a*d)^3*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4* (-(b*c) + a*d)*(-3*a*A*b^3*c^5 + 3*b^4*B*c^5 + 9*a^2*A*b^2*c^4*d - 9*A*b^4 *c^4*d - 9*a^3*A*b*c^3*d^2 + 15*a*A*b^3*c^3*d^2 - a^2*b^2*B*c^3*d^2 - 5*b^ 4*B*c^3*d^2 + 3*a^4*A*c^2*d^3 - 20*a^2*A*b^2*c^2*d^3 + 17*A*b^4*c^2*d^3 + 10*a^3*b*B*c^2*d^3 - 10*a*b^3*B*c^2*d^3 + 5*a^3*A*b*c*d^4 - 8*a*A*b^3*c*d^ 4 - 4*a^4*B*c*d^4 + 5*a^2*b^2*B*c*d^4 + 2*b^4*B*c*d^4 + a^4*A*d^5 + 7*a^2* A*b^2*d^5 - 8*A*b^4*d^5 - 6*a^3*b*B*d^5 + 6*a*b^3*B*d^5)*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*S qrt[((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)*(-3*A*b^4*c^5 + 3*a*b^3*B*c^5 - 3*a*A*b^3*c^4*d...
Time = 3.42 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3042, 3479, 27, 3042, 3534, 27, 3042, 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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3479 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {A d a^2-(A b c-3 b B d) a+2 b (A b-a B) d \sin ^2(e+f x)+b^2 (B c-4 A d)-(A b-a B) (b c+a d) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {A d a^2-(A b c-3 b B d) a+2 b (A b-a B) d \sin ^2(e+f x)+b^2 (B c-4 A d)-(A b-a B) (b c+a d) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {A d a^2-(A b c-3 b B d) a+2 b (A b-a B) d \sin (e+f x)^2+b^2 (B c-4 A d)-(A b-a B) (b c+a d) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 \int -\frac {3 d^2 (A c-B d) a^3-b d \left (6 A c^2-B d c-5 A d^2\right ) a^2+3 b^2 (A c-B d) \left (c^2-2 d^2\right ) a-b^3 \left (3 B c^3-9 A d c^2-2 B d^2 c+8 A d^3\right )+\left (B \left (c d^2 a^3-3 b d \left (2 c^2-d^2\right ) a^2-b^2 c \left (3 c^2-2 d^2\right ) a+3 b^3 c^2 d\right )+A \left (3 \left (c^3-2 c d^2\right ) b^3+a d \left (3 c^2-2 d^2\right ) b^2+3 a^2 c d^2 b-a^3 d^3\right )\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {3 d^2 (A c-B d) a^3-b d \left (6 A c^2-B d c-5 A d^2\right ) a^2+3 b^2 (A c-B d) \left (c^2-2 d^2\right ) a-b^3 \left (3 B c^3-9 A d c^2-2 B d^2 c+8 A d^3\right )+\left (B \left (c d^2 a^3-3 b d \left (2 c^2-d^2\right ) a^2-b^2 c \left (3 c^2-2 d^2\right ) a+3 b^3 c^2 d\right )+A \left (3 \left (c^3-2 c d^2\right ) b^3+a d \left (3 c^2-2 d^2\right ) b^2+3 a^2 c d^2 b-a^3 d^3\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {3 d^2 (A c-B d) a^3-b d \left (6 A c^2-B d c-5 A d^2\right ) a^2+3 b^2 (A c-B d) \left (c^2-2 d^2\right ) a-b^3 \left (3 B c^3-9 A d c^2-2 B d^2 c+8 A d^3\right )+\left (B \left (c d^2 a^3-3 b d \left (2 c^2-d^2\right ) a^2-b^2 c \left (3 c^2-2 d^2\right ) a+3 b^3 c^2 d\right )+A \left (3 \left (c^3-2 c d^2\right ) b^3+a d \left (3 c^2-2 d^2\right ) b^2+3 a^2 c d^2 b-a^3 d^3\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {(a-b) \left (B \left (a^2 d^2 (c+3 d)-6 a b d \left (c^2-d^2\right )+b^2 (-c) \left (3 c^2+3 c d-2 d^2\right )\right )-A \left (a^2 d^2 (3 c+d)-6 a b d \left (c^2-d^2\right )+b^2 \left (3 c^3-9 c^2 d-6 c d^2+8 d^3\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {\left (A \left (4 a^3 c d^3-a^2 b d^2 \left (9 c^2-5 d^2\right )-4 a b^2 c d^3-\left (b^3 \left (3 c^4-15 c^2 d^2+8 d^4\right )\right )\right )+B \left (a^3 \left (-d^2\right ) \left (c^2+3 d^2\right )+2 a^2 b c d \left (3 c^2-d^2\right )+a b^2 \left (3 c^4-5 c^2 d^2+6 d^4\right )-2 b^3 c d \left (3 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{3 \left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {(a-b) \left (B \left (a^2 d^2 (c+3 d)-6 a b d \left (c^2-d^2\right )+b^2 (-c) \left (3 c^2+3 c d-2 d^2\right )\right )-A \left (a^2 d^2 (3 c+d)-6 a b d \left (c^2-d^2\right )+b^2 \left (3 c^3-9 c^2 d-6 c d^2+8 d^3\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {\left (A \left (4 a^3 c d^3-a^2 b d^2 \left (9 c^2-5 d^2\right )-4 a b^2 c d^3-\left (b^3 \left (3 c^4-15 c^2 d^2+8 d^4\right )\right )\right )+B \left (a^3 \left (-d^2\right ) \left (c^2+3 d^2\right )+2 a^2 b c d \left (3 c^2-d^2\right )+a b^2 \left (3 c^4-5 c^2 d^2+6 d^4\right )-2 b^3 c d \left (3 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{3 \left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \cos (e+f x) \left (a^2 (-A) d^2+a^2 B c d+3 a b B \left (c^2-d^2\right )-A b^2 \left (3 c^2-4 d^2\right )-b^2 B c d\right ) \sqrt {a+b \sin (e+f x)}}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\left (A \left (4 a^3 c d^3-a^2 b d^2 \left (9 c^2-5 d^2\right )-4 a b^2 c d^3-\left (b^3 \left (3 c^4-15 c^2 d^2+8 d^4\right )\right )\right )+B \left (a^3 \left (-d^2\right ) \left (c^2+3 d^2\right )+2 a^2 b c d \left (3 c^2-d^2\right )+a b^2 \left (3 c^4-5 c^2 d^2+6 d^4\right )-2 b^3 c d \left (3 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}-\frac {2 (a-b) \sqrt {a+b} \sec (e+f x) \left (B \left (a^2 d^2 (c+3 d)-6 a b d \left (c^2-d^2\right )+b^2 (-c) \left (3 c^2+3 c d-2 d^2\right )\right )-A \left (a^2 d^2 (3 c+d)-6 a b d \left (c^2-d^2\right )+b^2 \left (3 c^3-9 c^2 d-6 c d^2+8 d^3\right )\right )\right ) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 b (A b-a B) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \left (-A d^2 a^2+B c d a^2+3 b B \left (c^2-d^2\right ) a-b^2 B c d-A b^2 \left (3 c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (B \left (-d^2 \left (c^2+3 d^2\right ) a^3+2 b c d \left (3 c^2-d^2\right ) a^2+b^2 \left (3 c^4-5 d^2 c^2+6 d^4\right ) a-2 b^3 c d \left (3 c^2-d^2\right )\right )+A \left (-\left (\left (3 c^4-15 d^2 c^2+8 d^4\right ) b^3\right )-4 a c d^3 b^2-a^2 d^2 \left (9 c^2-5 d^2\right ) b+4 a^3 c d^3\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d)^2 f}-\frac {2 (a-b) \sqrt {a+b} \left (B \left (-c \left (3 c^2+3 d c-2 d^2\right ) b^2-6 a d \left (c^2-d^2\right ) b+a^2 d^2 (c+3 d)\right )-A \left (\left (3 c^3-9 d c^2-6 d^2 c+8 d^3\right ) b^2-6 a d \left (c^2-d^2\right ) b+a^2 d^2 (3 c+d)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}}{3 (b c-a d) \left (c^2-d^2\right )}}{\left (a^2-b^2\right ) (b c-a d)}\) |
Input:
Int[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^ (5/2)),x]
Output:
(2*b*(A*b - a*B)*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - ((2*d*(a^2*B*c*d - b^2*B*c*d - a^2* A*d^2 - A*b^2*(3*c^2 - 4*d^2) + 3*a*b*B*(c^2 - d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - ((2*(a - b)*Sqrt[a + b]*(B*(2*a^2*b*c*d*(3*c^2 - d^2) - 2*b^3*c*d*(3*c^ 2 - d^2) - a^3*d^2*(c^2 + 3*d^2) + a*b^2*(3*c^4 - 5*c^2*d^2 + 6*d^4)) + A* (4*a^3*c*d^3 - 4*a*b^2*c*d^3 - a^2*b*d^2*(9*c^2 - 5*d^2) - b^3*(3*c^4 - 15 *c^2*d^2 + 8*d^4)))*EllipticE[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*S in[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]))/((c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) - (2*(a - b)*Sqrt[a + b]*(B*(a^2*d^2*(c + 3*d) - b^2*c*(3*c^2 + 3*c*d - 2* d^2) - 6*a*b*d*(c^2 - d^2)) - A*(a^2*d^2*(3*c + d) - 6*a*b*d*(c^2 - d^2) + b^2*(3*c^3 - 9*c^2*d - 6*c*d^2 + 8*d^3)))*EllipticF[ArcSin[(Sqrt[c + d]*S qrt[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]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f))/(3*(b*c - a*d)*(c^2 - d^2)))/((a^2 - b^2)*(b*c - ...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin [e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) *(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n }, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(I ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0]) ))
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(397568\) vs. \(2(814)=1628\).
Time = 48.46 (sec) , antiderivative size = 397569, normalized size of antiderivative = 463.37
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(397569\) |
| default | \(\text {Expression too large to display}\) | \(422919\) |
Input:
int((A+B*sin(f*x+e))/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x,metho d=_RETURNVERBOSE)
Output:
result too large to display
\[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x , algorithm="fricas")
Output:
integral((B*sin(f*x + e) + A)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(6*a*b*c^2*d + 2*a*b*d^3 + (3*b^2*c*d^2 + 2*a*b*d^3)*cos(f*x + e)^4 + (a^2 + b^2)*c^3 + 3*(a^2 + b^2)*c*d^2 - (b^2*c^3 + 6*a*b*c^2*d + 4*a*b*d ^3 + 3*(a^2 + 2*b^2)*c*d^2)*cos(f*x + e)^2 + (b^2*d^3*cos(f*x + e)^4 + 2*a *b*c^3 + 6*a*b*c*d^2 + 3*(a^2 + b^2)*c^2*d + (a^2 + b^2)*d^3 - (3*b^2*c^2* d + 6*a*b*c*d^2 + (a^2 + 2*b^2)*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sin(f*x+e))/(a+b*sin(f*x+e))**(3/2)/(c+d*sin(f*x+e))**(5/2) ,x)
Output:
Timed out
\[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x , algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)/((b*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^(5/2)), x)
\[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x , algorithm="giac")
Output:
integrate((B*sin(f*x + e) + A)/((b*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^(5/2)), x)
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int((A + B*sin(e + f*x))/((a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^ (5/2)),x)
Output:
int((A + B*sin(e + f*x))/((a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^ (5/2)), x)
\[ \int \frac {A+B \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}}{\sin \left (f x +e \right )^{4} b \,d^{3}+\sin \left (f x +e \right )^{3} a \,d^{3}+3 \sin \left (f x +e \right )^{3} b c \,d^{2}+3 \sin \left (f x +e \right )^{2} a c \,d^{2}+3 \sin \left (f x +e \right )^{2} b \,c^{2} d +3 \sin \left (f x +e \right ) a \,c^{2} d +\sin \left (f x +e \right ) b \,c^{3}+a \,c^{3}}d x \] Input:
int((A+B*sin(f*x+e))/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x)
Output:
int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a))/(sin(e + f*x)**4*b *d**3 + sin(e + f*x)**3*a*d**3 + 3*sin(e + f*x)**3*b*c*d**2 + 3*sin(e + f* x)**2*a*c*d**2 + 3*sin(e + f*x)**2*b*c**2*d + 3*sin(e + f*x)*a*c**2*d + si n(e + f*x)*b*c**3 + a*c**3),x)