Integrand size = 35, antiderivative size = 508 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=-\frac {d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^3 (c-d)^5 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )+A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac {(2 A c+3 B c-11 A d+6 B d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\left (3 B \left (c^2-10 c d-12 d^2\right )+A \left (2 c^2-15 c d+76 d^2\right )\right ) \cos (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^2}-\frac {d \left (3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )+A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))} \] Output:
-d^2*(A*d*(20*c^2+30*c*d+13*d^2)-3*B*(4*c^3+8*c^2*d+7*c*d^2+2*d^3))*arctan ((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/a^3/(c-d)^5/(c+d)^2/(c^2-d^2)^( 1/2)/f-1/30*d*(3*B*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)+A*(4*c^3-30*c^2*d+146* c*d^2+195*d^3))*cos(f*x+e)/a^3/(c-d)^4/(c+d)/f/(c+d*sin(f*x+e))^2-1/5*(A-B )*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^2-1/15*(2*A*c-11* A*d+3*B*c+6*B*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e) )^2-1/15*(3*B*(c^2-10*c*d-12*d^2)+A*(2*c^2-15*c*d+76*d^2))*cos(f*x+e)/(c-d )^3/f/(a^3+a^3*sin(f*x+e))/(c+d*sin(f*x+e))^2-1/30*d*(3*B*(2*c^4-20*c^3*d- 119*c^2*d^2-130*c*d^3-48*d^4)+A*(4*c^4-30*c^3*d+142*c^2*d^2+525*c*d^3+304* d^4))*cos(f*x+e)/a^3/(c-d)^5/(c+d)^2/f/(c+d*sin(f*x+e))
Time = 11.78 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (12 (A-B) (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )+6 (-A+B) (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 (c-d) (A (2 c-17 d)+3 B (c+4 d)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+4 \left (3 B \left (c^2-12 c d-19 d^2\right )+A \left (2 c^2-19 c d+107 d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+\frac {30 d^2 \left (-A d \left (20 c^2+30 c d+13 d^2\right )+3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d)^2 \sqrt {c^2-d^2}}+\frac {15 (c-d) d^3 (B c-A d) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d) (c+d \sin (e+f x))^2}+\frac {15 d^3 \left (-3 A d (3 c+2 d)+B \left (7 c^2+6 c d+2 d^2\right )\right ) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{(c+d)^2 (c+d \sin (e+f x))}\right )}{30 a^3 (c-d)^5 f (1+\sin (e+f x))^3} \] Input:
Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x] )^3),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*(A - B)*(c - d)^2*Sin[(e + f*x) /2] + 6*(-A + B)*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + S in[(e + f*x)/2])^2 - 2*(c - d)*(A*(2*c - 17*d) + 3*B*(c + 4*d))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 4*(3*B*(c^2 - 12*c*d - 19*d^2) + A*(2*c^2 - 19*c*d + 107*d^2))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2] )^4 + (30*d^2*(-(A*d*(20*c^2 + 30*c*d + 13*d^2)) + 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)^2*Sqrt[c^2 - d^2]) + (15*(c - d )*d^3*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(( c + d)*(c + d*Sin[e + f*x])^2) + (15*d^3*(-3*A*d*(3*c + 2*d) + B*(7*c^2 + 6*c*d + 2*d^2))*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/((c + d)^2*(c + d*Sin[e + f*x]))))/(30*a^3*(c - d)^5*f*(1 + Sin[e + f*x])^3)
Time = 2.35 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 3457, 25, 3042, 3457, 25, 3042, 3457, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle -\frac {\int -\frac {a (2 A c+3 B c-7 A d+2 B d)+4 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (2 A c+3 B c-7 A d+2 B d)+4 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (2 A c+3 B c-7 A d+2 B d)+4 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (3 B \left (c^2-7 d c-6 d^2\right )+A \left (2 c^2-9 d c+43 d^2\right )\right ) a^2+3 d (2 A c+3 B c-11 A d+6 B d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 B \left (c^2-7 d c-6 d^2\right )+A \left (2 c^2-9 d c+43 d^2\right )\right ) a^2+3 d (2 A c+3 B c-11 A d+6 B d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 B \left (c^2-7 d c-6 d^2\right )+A \left (2 c^2-9 d c+43 d^2\right )\right ) a^2+3 d (2 A c+3 B c-11 A d+6 B d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 d^2 (2 A c+33 B c-65 A d+30 B d) a^3+2 d \left (3 B \left (c^2-10 d c-12 d^2\right )+A \left (2 c^2-15 d c+76 d^2\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 d^2 (2 A c+33 B c-65 A d+30 B d) a^3+2 d \left (3 B \left (c^2-10 d c-12 d^2\right )+A \left (2 c^2-15 d c+76 d^2\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 d^2 (2 A c+33 B c-65 A d+30 B d) a^3+2 d \left (3 B \left (c^2-10 d c-12 d^2\right )+A \left (2 c^2-15 d c+76 d^2\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {2 d^2 \left (2 A c^2+93 B c^2-165 A d c+150 B d c-152 A d^2+72 B d^2\right ) a^3+d \left (3 B \left (2 c^3-20 d c^2-57 d^2 c-30 d^3\right )+A \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 d^2 \left (2 A c^2+93 B c^2-165 A d c+150 B d c-152 A d^2+72 B d^2\right ) a^3+d \left (3 B \left (2 c^3-20 d c^2-57 d^2 c-30 d^3\right )+A \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 d^2 \left (2 A c^2+93 B c^2-165 A d c+150 B d c-152 A d^2+72 B d^2\right ) a^3+d \left (3 B \left (2 c^3-20 d c^2-57 d^2 c-30 d^3\right )+A \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int \frac {15 a^3 d^2 \left (A d \left (20 c^2+30 d c+13 d^2\right )-3 B \left (4 c^3+8 d c^2+7 d^2 c+2 d^3\right )\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {30 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {60 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {30 a^3 d^2 \left (A d \left (20 c^2+30 c d+13 d^2\right )-3 B \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {a^3 d \left (A \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right )+3 B \left (2 c^4-20 c^3 d-119 c^2 d^2-130 c d^3-48 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (A \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+3 B \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (A \left (2 c^2-15 c d+76 d^2\right )+3 B \left (c^2-10 c d-12 d^2\right )\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 A c-11 A d+3 B c+6 B d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
Input:
Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x ]
Output:
-1/5*((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) + (-1/3*(a*(2*A*c + 3*B*c - 11*A*d + 6*B*d)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (-((a^2*(3*B*(c^2 - 10*c*d - 12*d^2) + A*(2*c^2 - 15*c*d + 76*d^2))*Cos[e + f*x])/((c - d)*f *(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)) + (-1/2*(a^3*d*(3*B*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3) + A*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^ 3))*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((-30*a^3*d^2*( A*d*(20*c^2 + 30*c*d + 13*d^2) - 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))* ArcTan[(2*d + 2*c*Tan[(e + f*x)/2])/(2*Sqrt[c^2 - d^2])])/((c^2 - d^2)^(3/ 2)*f) - (a^3*d*(3*B*(2*c^4 - 20*c^3*d - 119*c^2*d^2 - 130*c*d^3 - 48*d^4) + A*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4))*Cos[e + f*x])/ ((c^2 - d^2)*f*(c + d*Sin[e + f*x])))/(2*(c^2 - d^2)))/(a^2*(c - d)))/(3*a ^2*(c - d)))/(5*a^2*(c - d))
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_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Time = 5.00 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (11 A \,c^{2} d +6 A c \,d^{2}-2 A \,d^{3}-9 B \,c^{3}-6 B \,c^{2} d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 A \,c^{4} d +6 A \,c^{3} d^{2}+19 A \,c^{2} d^{3}+12 A c \,d^{4}-2 A \,d^{5}-8 B \,c^{5}-6 B \,c^{4} d -17 B \,c^{3} d^{2}-12 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (29 A \,c^{2} d +18 A c \,d^{2}-2 A \,d^{3}-23 B \,c^{3}-18 B \,c^{2} d -4 B c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 A \,c^{2} d +6 A c \,d^{2}-A \,d^{3}-8 B \,c^{3}-6 B \,c^{2} d -B c \,d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (20 A \,c^{2} d +30 A c \,d^{2}+13 A \,d^{3}-12 B \,c^{3}-24 B \,c^{2} d -21 B c \,d^{2}-6 B \,d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{5}}-\frac {-8 A +8 B}{2 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A -4 B \right )}{5 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-4 A c +10 A d +2 B c -8 B d}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 A c -14 A d -6 B c +12 B d \right )}{3 \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (A \,c^{2}-5 A c d +10 A \,d^{2}-6 B \,d^{2}\right )}{\left (c -d \right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} f}\) | \(639\) |
| default | \(\frac {-\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (11 A \,c^{2} d +6 A c \,d^{2}-2 A \,d^{3}-9 B \,c^{3}-6 B \,c^{2} d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 A \,c^{4} d +6 A \,c^{3} d^{2}+19 A \,c^{2} d^{3}+12 A c \,d^{4}-2 A \,d^{5}-8 B \,c^{5}-6 B \,c^{4} d -17 B \,c^{3} d^{2}-12 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (29 A \,c^{2} d +18 A c \,d^{2}-2 A \,d^{3}-23 B \,c^{3}-18 B \,c^{2} d -4 B c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 A \,c^{2} d +6 A c \,d^{2}-A \,d^{3}-8 B \,c^{3}-6 B \,c^{2} d -B c \,d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (20 A \,c^{2} d +30 A c \,d^{2}+13 A \,d^{3}-12 B \,c^{3}-24 B \,c^{2} d -21 B c \,d^{2}-6 B \,d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{5}}-\frac {-8 A +8 B}{2 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A -4 B \right )}{5 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-4 A c +10 A d +2 B c -8 B d}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 A c -14 A d -6 B c +12 B d \right )}{3 \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (A \,c^{2}-5 A c d +10 A \,d^{2}-6 B \,d^{2}\right )}{\left (c -d \right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} f}\) | \(639\) |
| risch | \(\text {Expression too large to display}\) | \(2987\) |
Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x,method=_RETUR NVERBOSE)
Output:
2/f/a^3*(-d^2/(c-d)^5*((1/2*d^2*(11*A*c^2*d+6*A*c*d^2-2*A*d^3-9*B*c^3-6*B* c^2*d)/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*d*(10*A*c^4*d+6*A*c^3*d^ 2+19*A*c^2*d^3+12*A*c*d^4-2*A*d^5-8*B*c^5-6*B*c^4*d-17*B*c^3*d^2-12*B*c^2* d^3-2*B*c*d^4)/(c^2+2*c*d+d^2)/c^2*tan(1/2*f*x+1/2*e)^2+1/2*d^2*(29*A*c^2* d+18*A*c*d^2-2*A*d^3-23*B*c^3-18*B*c^2*d-4*B*c*d^2)/c/(c^2+2*c*d+d^2)*tan( 1/2*f*x+1/2*e)+1/2*d*(10*A*c^2*d+6*A*c*d^2-A*d^3-8*B*c^3-6*B*c^2*d-B*c*d^2 )/(c^2+2*c*d+d^2))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2 *(20*A*c^2*d+30*A*c*d^2+13*A*d^3-12*B*c^3-24*B*c^2*d-21*B*c*d^2-6*B*d^3)/( c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^ 2-d^2)^(1/2)))-1/4*(-8*A+8*B)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^4-1/5*(4*A-4* B)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^5-1/2*(-4*A*c+10*A*d+2*B*c-8*B*d)/(c-d)^ 4/(tan(1/2*f*x+1/2*e)+1)^2-1/3*(8*A*c-14*A*d-6*B*c+12*B*d)/(c-d)^4/(tan(1/ 2*f*x+1/2*e)+1)^3-(A*c^2-5*A*c*d+10*A*d^2-6*B*d^2)/(c-d)^5/(tan(1/2*f*x+1/ 2*e)+1))
Leaf count of result is larger than twice the leaf count of optimal. 3599 vs. \(2 (493) = 986\).
Time = 0.37 (sec) , antiderivative size = 7283, normalized size of antiderivative = 14.34 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algori thm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algori thm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1224 vs. \(2 (493) = 986\).
Time = 0.36 (sec) , antiderivative size = 1224, normalized size of antiderivative = 2.41 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algori thm="giac")
Output:
1/15*(15*(12*B*c^3*d^2 - 20*A*c^2*d^3 + 24*B*c^2*d^3 - 30*A*c*d^4 + 21*B*c *d^4 - 13*A*d^5 + 6*B*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arct an((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((a^3*c^7 - 3*a^3*c^6*d + a^3*c^5*d^2 + 5*a^3*c^4*d^3 - 5*a^3*c^3*d^4 - a^3*c^2*d^5 + 3*a^3*c*d^6 - a^3*d^7)*sqrt(c^2 - d^2)) + 15*(9*B*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 11* A*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 6*B*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 6* A*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 2*A*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*B* c^5*d^3*tan(1/2*f*x + 1/2*e)^2 - 10*A*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 6*B *c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 6*A*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 17* B*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 - 19*A*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 1 2*B*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 - 12*A*c*d^7*tan(1/2*f*x + 1/2*e)^2 + 2 *B*c*d^7*tan(1/2*f*x + 1/2*e)^2 + 2*A*d^8*tan(1/2*f*x + 1/2*e)^2 + 23*B*c^ 4*d^4*tan(1/2*f*x + 1/2*e) - 29*A*c^3*d^5*tan(1/2*f*x + 1/2*e) + 18*B*c^3* d^5*tan(1/2*f*x + 1/2*e) - 18*A*c^2*d^6*tan(1/2*f*x + 1/2*e) + 4*B*c^2*d^6 *tan(1/2*f*x + 1/2*e) + 2*A*c*d^7*tan(1/2*f*x + 1/2*e) + 8*B*c^5*d^3 - 10* A*c^4*d^4 + 6*B*c^4*d^4 - 6*A*c^3*d^5 + B*c^3*d^5 + A*c^2*d^6)/((a^3*c^9 - 3*a^3*c^8*d + a^3*c^7*d^2 + 5*a^3*c^6*d^3 - 5*a^3*c^5*d^4 - a^3*c^4*d^5 + 3*a^3*c^3*d^6 - a^3*c^2*d^7)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2) - 2*(15*A*c^2*tan(1/2*f*x + 1/2*e)^4 - 75*A*c*d*tan(1/2*f *x + 1/2*e)^4 + 150*A*d^2*tan(1/2*f*x + 1/2*e)^4 - 90*B*d^2*tan(1/2*f*x...
Time = 45.30 (sec) , antiderivative size = 2387, normalized size of antiderivative = 4.70 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^3),x )
Output:
((15*A*d^6 - 14*A*c^6 - 6*B*c^6 - 404*A*c^2*d^4 - 420*A*c^3*d^3 - 92*A*c^4 *d^2 + 234*B*c^2*d^4 + 450*B*c^3*d^3 + 222*B*c^4*d^2 - 90*A*c*d^5 + 60*A*c ^5*d + 15*B*c*d^5 + 30*B*c^5*d)/(15*(c + d)^2*(c - d)*(c^4 - 4*c^3*d - 4*c *d^3 + d^4 + 6*c^2*d^2)) + (tan(e/2 + (f*x)/2)^7*(2*A*d^8 - 4*A*c^8 - 2*B* c^8 - 49*A*c^2*d^6 - 141*A*c^3*d^5 - 200*A*c^4*d^4 - 122*A*c^5*d^3 + 2*A*c ^6*d^2 + 12*B*c^2*d^6 + 95*B*c^3*d^5 + 187*B*c^4*d^4 + 146*B*c^5*d^3 + 58* B*c^6*d^2 - 2*A*c*d^7 + 10*A*c^7*d + 2*B*c*d^7 + 6*B*c^7*d))/(c^2*(c - d)* (2*c*d + c^2 + d^2)*(c^4 - 4*c^3*d - 4*c*d^3 + d^4 + 6*c^2*d^2)) + (tan(e/ 2 + (f*x)/2)^6*(30*A*d^8 - 28*A*c^8 - 6*B*c^8 - 759*A*c^2*d^6 - 1707*A*c^3 *d^5 - 1960*A*c^4*d^4 - 870*A*c^5*d^3 + 62*A*c^6*d^2 + 336*B*c^2*d^6 + 125 7*B*c^3*d^5 + 1893*B*c^4*d^4 + 1350*B*c^5*d^3 + 414*B*c^6*d^2 - 114*A*c*d^ 7 + 54*A*c^7*d + 30*B*c*d^7 + 18*B*c^7*d))/(3*c^2*(c - d)*(2*c*d + c^2 + d ^2)*(c^4 - 4*c^3*d - 4*c*d^3 + d^4 + 6*c^2*d^2)) + (tan(e/2 + (f*x)/2)^5*( 60*A*d^8 - 32*A*c^8 - 18*B*c^8 - 1857*A*c^2*d^6 - 3763*A*c^3*d^5 - 3560*A* c^4*d^4 - 1294*A*c^5*d^3 + 70*A*c^6*d^2 + 900*B*c^2*d^6 + 2859*B*c^3*d^5 + 3705*B*c^4*d^4 + 2358*B*c^5*d^3 + 678*B*c^6*d^2 - 270*A*c*d^7 + 62*A*c^7* d + 60*B*c*d^7 + 42*B*c^7*d))/(3*c^2*(c - d)*(2*c*d + c^2 + d^2)*(c^4 - 4* c^3*d - 4*c*d^3 + d^4 + 6*c^2*d^2)) + (tan(e/2 + (f*x)/2)^2*(30*A*d^8 - 10 8*A*c^8 - 42*B*c^8 - 2501*A*c^2*d^6 - 8725*A*c^3*d^5 - 10616*A*c^4*d^4 - 4 810*A*c^5*d^3 + 10*A*c^6*d^2 + 1056*B*c^2*d^6 + 5235*B*c^3*d^5 + 9891*B...
Time = 3.21 (sec) , antiderivative size = 12007, normalized size of antiderivative = 23.64 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x)
Output:
( - 1500*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2) )*tan((e + f*x)/2)**9*a*c**6*d**3 - 3450*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9*a*c**5*d**4 - 2775*s qrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9*a*c**4*d**5 - 780*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9*a*c**3*d**6 + 900*sqrt(c**2 - d **2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9* b*c**7*d**2 + 2520*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c* *2 - d**2))*tan((e + f*x)/2)**9*b*c**6*d**3 + 3015*sqrt(c**2 - d**2)*atan( (tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9*b*c**5*d** 4 + 1710*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2) )*tan((e + f*x)/2)**9*b*c**4*d**5 + 360*sqrt(c**2 - d**2)*atan((tan((e + f *x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**9*b*c**3*d**6 - 7500*sq rt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**8*a*c**6*d**3 - 23250*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**8*a*c**5*d**4 - 27675*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)* *8*a*c**4*d**5 - 15000*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqr t(c**2 - d**2))*tan((e + f*x)/2)**8*a*c**3*d**6 - 3120*sqrt(c**2 - d**2)*a tan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*tan((e + f*x)/2)**8*a*c...