Integrand size = 35, antiderivative size = 386 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\frac {d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a^2 (c-d)^4 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (A \left (2 c^2-16 c d-21 d^2\right )+B \left (4 c^2+19 c d+12 d^2\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^2}-\frac {(A c+2 B c-8 A d+5 B d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{6 a^2 (c-d)^4 (c+d)^2 f (c+d \sin (e+f x))} \]
-1/6*d*(A*(2*c^2-16*c*d-21*d^2)+B*(4*c^2+19*c*d+12*d^2))*cos(f*x+e)/a^2/(c -d)^3/(c+d)/f/(c+d*sin(f*x+e))^2-1/3*(A*c-8*A*d+2*B*c+5*B*d)*cos(f*x+e)/a^ 2/(c-d)^2/f/(1+sin(f*x+e))/(c+d*sin(f*x+e))^2-1/3*(A-B)*cos(f*x+e)/(c-d)/f /(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^2-1/6*d*(A*(2*c^3-16*c^2*d-59*c*d^2-3 2*d^3)+B*(4*c^3+37*c^2*d+44*c*d^2+20*d^3))*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/ f/(c+d*sin(f*x+e))+d*(A*d*(12*c^2+16*c*d+7*d^2)-B*(6*c^3+12*c^2*d+13*c*d^2 +4*d^3))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/a^2/(c-d)^4/(c+d )^2/f/(c^2-d^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1257\) vs. \(2(386)=772\).
Time = 10.90 (sec) , antiderivative size = 1257, normalized size of antiderivative = 3.26 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-48*d*(-(A*d*(12*c^2 + 16*c*d + 7 *d^2)) + B*(6*c^3 + 12*c^2*d + 13*c*d^2 + 4*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])^3)/Sqrt[c^2 - d^2] + ((-(A*d*(96*c^4 + 524*c^3*d + 776*c^2*d^2 + 487*c*d^3 + 112*d^4) ) + B*(48*c^5 + 240*c^4*d + 536*c^3*d^2 + 701*c^2*d^3 + 400*c*d^4 + 70*d^5 ))*Cos[(e + f*x)/2] - (A*(16*c^5 - 80*c^4*d - 536*c^3*d^2 - 1028*c^2*d^3 - 695*c*d^4 - 134*d^5) + B*(32*c^5 + 224*c^4*d + 728*c^3*d^2 + 893*c^2*d^3 + 482*c*d^4 + 98*d^5))*Cos[(3*(e + f*x))/2] + 24*B*c^3*d^2*Cos[(5*(e + f*x ))/2] - 12*A*c^2*d^3*Cos[(5*(e + f*x))/2] + 21*B*c^2*d^3*Cos[(5*(e + f*x)) /2] - 15*A*c*d^4*Cos[(5*(e + f*x))/2] - 18*B*c*d^4*Cos[(5*(e + f*x))/2] + 6*A*d^5*Cos[(5*(e + f*x))/2] - 6*B*d^5*Cos[(5*(e + f*x))/2] + 4*A*c^3*d^2* Cos[(7*(e + f*x))/2] + 8*B*c^3*d^2*Cos[(7*(e + f*x))/2] - 32*A*c^2*d^3*Cos [(7*(e + f*x))/2] + 59*B*c^2*d^3*Cos[(7*(e + f*x))/2] - 97*A*c*d^4*Cos[(7* (e + f*x))/2] + 76*B*c*d^4*Cos[(7*(e + f*x))/2] - 52*A*d^5*Cos[(7*(e + f*x ))/2] + 34*B*d^5*Cos[(7*(e + f*x))/2] + 48*A*c^5*Sin[(e + f*x)/2] + 48*B*c ^5*Sin[(e + f*x)/2] - 224*A*c^4*d*Sin[(e + f*x)/2] + 416*B*c^4*d*Sin[(e + f*x)/2] - 872*A*c^3*d^2*Sin[(e + f*x)/2] + 992*B*c^3*d^2*Sin[(e + f*x)/2] - 1144*A*c^2*d^3*Sin[(e + f*x)/2] + 967*B*c^2*d^3*Sin[(e + f*x)/2] - 685*A *c*d^4*Sin[(e + f*x)/2] + 496*B*c*d^4*Sin[(e + f*x)/2] - 168*A*d^5*Sin[(e + f*x)/2] + 126*B*d^5*Sin[(e + f*x)/2] + 48*B*c^4*d*Sin[(3*(e + f*x))/2...
Time = 1.65 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3457, 25, 3042, 3457, 3042, 3233, 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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle -\frac {\int -\frac {a (A (c-5 d)+2 B (c+d))+3 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (A (c-5 d)+2 B (c+d))+3 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (A (c-5 d)+2 B (c+d))+3 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int \frac {3 a^2 d (3 B c-7 A d+4 B d)-2 a^2 d (A c+2 B c-8 A d+5 B d) \sin (e+f x)}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {3 a^2 d (3 B c-7 A d+4 B d)-2 a^2 d (A c+2 B c-8 A d+5 B d) \sin (e+f x)}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\int \frac {2 d \left (A d (19 c+16 d)-B \left (9 c^2+16 d c+10 d^2\right )\right ) a^2+d \left (A \left (2 c^2-16 d c-21 d^2\right )+B \left (4 c^2+19 d c+12 d^2\right )\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\int \frac {2 d \left (A d (19 c+16 d)-B \left (9 c^2+16 d c+10 d^2\right )\right ) a^2+d \left (A \left (2 c^2-16 d c-21 d^2\right )+B \left (4 c^2+19 d c+12 d^2\right )\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {-\frac {\int -\frac {3 a^2 d \left (A d \left (12 c^2+16 d c+7 d^2\right )-B \left (6 c^3+12 d c^2+13 d^2 c+4 d^3\right )\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\frac {3 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\frac {3 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\frac {6 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {-\frac {12 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {a^2 d (3 d (-7 A d+3 B c+4 B d)+2 c (A c-8 A d+2 B c+5 B d)) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {\frac {6 a^2 d \left (A d \left (12 c^2+16 c d+7 d^2\right )-B \left (6 c^3+12 c^2 d+13 c d^2+4 d^3\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {a^2 d \left (A \left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right )+B \left (4 c^3+37 c^2 d+44 c d^2+20 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}}{a^2 (c-d)}-\frac {(A c-8 A d+2 B c+5 B d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}\) |
-1/3*((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (-(((A*c + 2*B*c - 8*A*d + 5*B*d)*Cos[e + f*x])/((c - d)*f*( 1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2)) - ((a^2*d*(3*d*(3*B*c - 7*A*d + 4*B*d) + 2*c*(A*c + 2*B*c - 8*A*d + 5*B*d))*Cos[e + f*x])/(2*(c^2 - d^2)* f*(c + d*Sin[e + f*x])^2) - ((6*a^2*d*(A*d*(12*c^2 + 16*c*d + 7*d^2) - B*( 6*c^3 + 12*c^2*d + 13*c*d^2 + 4*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^2*d*(A*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3) + B*(4*c^3 + 37*c^2*d + 44*c*d^2 + 20*d^3))*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))
3.3.78.3.1 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_.)*(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 = 4.52 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-2 B +2 A \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A c -4 d A +3 d B \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d \left (\frac {\frac {d^{2} \left (9 c^{2} d A +4 d^{2} c A -2 A \,d^{3}-7 B \,c^{3}-4 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 A \,c^{4} d +4 A \,c^{3} d^{2}+15 A \,c^{2} d^{3}+8 A c \,d^{4}-2 A \,d^{5}-6 B \,c^{5}-4 B \,c^{4} d -13 B \,c^{3} d^{2}-8 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (23 c^{2} d A +12 d^{2} c A -2 A \,d^{3}-17 B \,c^{3}-12 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{2} d A +4 d^{2} c A -A \,d^{3}-6 B \,c^{3}-4 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2} d A +16 d^{2} c A +7 A \,d^{3}-6 B \,c^{3}-12 c^{2} d B -13 d^{2} c B -4 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\) | \(547\) |
| default | \(\frac {-\frac {2 \left (-2 B +2 A \right )}{3 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A c -4 d A +3 d B \right )}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d \left (\frac {\frac {d^{2} \left (9 c^{2} d A +4 d^{2} c A -2 A \,d^{3}-7 B \,c^{3}-4 c^{2} d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 A \,c^{4} d +4 A \,c^{3} d^{2}+15 A \,c^{2} d^{3}+8 A c \,d^{4}-2 A \,d^{5}-6 B \,c^{5}-4 B \,c^{4} d -13 B \,c^{3} d^{2}-8 B \,c^{2} d^{3}-2 B c \,d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}+\frac {d^{2} \left (23 c^{2} d A +12 d^{2} c A -2 A \,d^{3}-17 B \,c^{3}-12 c^{2} d B -4 d^{2} c B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c^{2} d A +4 d^{2} c A -A \,d^{3}-6 B \,c^{3}-4 c^{2} d B -d^{2} c B \right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (12 c^{2} d A +16 d^{2} c A +7 A \,d^{3}-6 B \,c^{3}-12 c^{2} d B -13 d^{2} c B -4 d^{3} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{4}}}{a^{2} f}\) | \(547\) |
| risch | \(\text {Expression too large to display}\) | \(2375\) |
2/f/a^2*(-1/3*(-2*B+2*A)/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^3-1/2*(2*B-2*A)/(c -d)^3/(tan(1/2*f*x+1/2*e)+1)^2-(A*c-4*A*d+3*B*d)/(c-d)^4/(tan(1/2*f*x+1/2* e)+1)+d/(c-d)^4*((1/2*d^2*(9*A*c^2*d+4*A*c*d^2-2*A*d^3-7*B*c^3-4*B*c^2*d)/ c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*d*(8*A*c^4*d+4*A*c^3*d^2+15*A*c ^2*d^3+8*A*c*d^4-2*A*d^5-6*B*c^5-4*B*c^4*d-13*B*c^3*d^2-8*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*(23*A*c^2*d+12*A*c*d ^2-2*A*d^3-17*B*c^3-12*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*(8*A*c^2*d+4*A*c*d^2-A*d^3-6*B*c^3-4*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*(12*A*c^2* d+16*A*c*d^2+7*A*d^3-6*B*c^3-12*B*c^2*d-13*B*c*d^2-4*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)) ))
Leaf count of result is larger than twice the leaf count of optimal. 2456 vs. \(2 (373) = 746\).
Time = 0.45 (sec) , antiderivative size = 4997, normalized size of antiderivative = 12.95 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
[-1/12*(4*(A - B)*c^7 - 4*(A - B)*c^6*d - 12*(A - B)*c^5*d^2 + 12*(A - B)* c^4*d^3 + 12*(A - B)*c^3*d^4 - 12*(A - B)*c^2*d^5 - 4*(A - B)*c*d^6 + 4*(A - B)*d^7 - 2*(2*(A + 2*B)*c^5*d^2 - (16*A - 37*B)*c^4*d^3 - (61*A - 40*B) *c^3*d^4 - (16*A + 17*B)*c^2*d^5 + (59*A - 44*B)*c*d^6 + 4*(8*A - 5*B)*d^7 )*cos(f*x + e)^4 - 2*(4*(A + 2*B)*c^6*d - 4*(7*A - 16*B)*c^5*d^2 - 118*(A - B)*c^4*d^3 - (106*A - 25*B)*c^3*d^4 + (71*A - 98*B)*c^2*d^5 + (134*A - 8 9*B)*c*d^6 + (43*A - 28*B)*d^7)*cos(f*x + e)^3 + 2*(2*(A + 2*B)*c^7 - 6*(2 *A - 3*B)*c^6*d - 12*(3*A - 4*B)*c^5*d^2 - 3*(18*A - 17*B)*c^4*d^3 - 3*(13 *A + B)*c^3*d^4 + 3*(13*A - 17*B)*c^2*d^5 + (73*A - 49*B)*c*d^6 + 9*(3*A - 2*B)*d^7)*cos(f*x + e)^2 + 3*(12*B*c^5*d - 24*(A - 2*B)*c^4*d^2 - 2*(40*A - 43*B)*c^3*d^3 - 6*(17*A - 14*B)*c^2*d^4 - 6*(10*A - 7*B)*c*d^5 - 2*(7*A - 4*B)*d^6 + (6*B*c^3*d^3 - 12*(A - B)*c^2*d^4 - (16*A - 13*B)*c*d^5 - (7 *A - 4*B)*d^6)*cos(f*x + e)^4 - (12*B*c^4*d^2 - 6*(4*A - 5*B)*c^3*d^3 - 2* (22*A - 19*B)*c^2*d^4 - 3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e)^3 - (6*B*c^5*d - 12*(A - 3*B)*c^4*d^2 - (64*A - 79*B)*c^3*d^3 - (107*A - 92*B)*c^2*d^4 - (76*A - 55*B)*c*d^5 - 3*(7*A - 4*B)*d^6)*cos(f*x + e)^2 + (6*B*c^5*d - 12*(A - 2*B)*c^4*d^2 - (40*A - 43*B)*c^3*d^3 - 3*(17*A - 14 *B)*c^2*d^4 - 3*(10*A - 7*B)*c*d^5 - (7*A - 4*B)*d^6)*cos(f*x + e) + (12*B *c^5*d - 24*(A - 2*B)*c^4*d^2 - 2*(40*A - 43*B)*c^3*d^3 - 6*(17*A - 14*B)* c^2*d^4 - 6*(10*A - 7*B)*c*d^5 - 2*(7*A - 4*B)*d^6 - (6*B*c^3*d^3 - 12*...
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 911 vs. \(2 (373) = 746\).
Time = 0.37 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.36 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
-1/3*(3*(6*B*c^3*d - 12*A*c^2*d^2 + 12*B*c^2*d^2 - 16*A*c*d^3 + 13*B*c*d^3 - 7*A*d^4 + 4*B*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c *tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((a^2*c^6 - 2*a^2*c^5*d - a^2 *c^4*d^2 + 4*a^2*c^3*d^3 - a^2*c^2*d^4 - 2*a^2*c*d^5 + a^2*d^6)*sqrt(c^2 - d^2)) + 3*(7*B*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 - 9*A*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*B*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 - 4*A*c^2*d^5*tan(1/2*f*x + 1/2*e)^3 + 2*A*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 6*B*c^5*d^2*tan(1/2*f*x + 1 /2*e)^2 - 8*A*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 4*B*c^4*d^3*tan(1/2*f*x + 1 /2*e)^2 - 4*A*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 + 13*B*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 - 15*A*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 8*B*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 8*A*c*d^6*tan(1/2*f*x + 1/2*e)^2 + 2*B*c*d^6*tan(1/2*f*x + 1/2 *e)^2 + 2*A*d^7*tan(1/2*f*x + 1/2*e)^2 + 17*B*c^4*d^3*tan(1/2*f*x + 1/2*e) - 23*A*c^3*d^4*tan(1/2*f*x + 1/2*e) + 12*B*c^3*d^4*tan(1/2*f*x + 1/2*e) - 12*A*c^2*d^5*tan(1/2*f*x + 1/2*e) + 4*B*c^2*d^5*tan(1/2*f*x + 1/2*e) + 2* A*c*d^6*tan(1/2*f*x + 1/2*e) + 6*B*c^5*d^2 - 8*A*c^4*d^3 + 4*B*c^4*d^3 - 4 *A*c^3*d^4 + B*c^3*d^4 + A*c^2*d^5)/((a^2*c^8 - 2*a^2*c^7*d - a^2*c^6*d^2 + 4*a^2*c^5*d^3 - a^2*c^4*d^4 - 2*a^2*c^3*d^5 + a^2*c^2*d^6)*(c*tan(1/2*f* x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2) + 2*(3*A*c*tan(1/2*f*x + 1 /2*e)^2 - 12*A*d*tan(1/2*f*x + 1/2*e)^2 + 9*B*d*tan(1/2*f*x + 1/2*e)^2 + 3 *A*c*tan(1/2*f*x + 1/2*e) + 3*B*c*tan(1/2*f*x + 1/2*e) - 21*A*d*tan(1/2...
Time = 17.88 (sec) , antiderivative size = 1686, normalized size of antiderivative = 4.37 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
(d*atan(((d*(4*a^2*c*d^6 - 2*a^2*d^7 - 2*a^2*c^6*d + 2*a^2*c^2*d^5 - 8*a^2 *c^3*d^4 + 2*a^2*c^4*d^3 + 4*a^2*c^5*d^2)*(6*B*c^3 - 7*A*d^3 + 4*B*d^3 - 1 6*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(2*a^2*(c + d)^(5/2)*(c - d)^(9/2)) + (c*d*tan(e/2 + (f*x)/2)*(2*a^2*c*d^5 - a^2*d^6 - a^2*c^6 + 2*a^2*c^5*d + a^2*c^2*d^4 - 4*a^2*c^3*d^3 + a^2*c^4*d^2)*(6*B*c^3 - 7*A*d^ 3 + 4*B*d^3 - 16*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(a^2*(c + d)^(5/2)*(c - d)^(9/2)))/(4*B*d^4 - 7*A*d^4 - 12*A*c^2*d^2 + 12*B*c^2*d^ 2 - 16*A*c*d^3 + 13*B*c*d^3 + 6*B*c^3*d))*(6*B*c^3 - 7*A*d^3 + 4*B*d^3 - 1 6*A*c*d^2 - 12*A*c^2*d + 13*B*c*d^2 + 12*B*c^2*d))/(a^2*f*(c + d)^(5/2)*(c - d)^(9/2)) - ((tan(e/2 + (f*x)/2)^5*(2*A*c^6 + 2*A*d^6 + 2*B*c^6 - 23*A* c^2*d^4 - 40*A*c^3*d^3 - 38*A*c^4*d^2 + 6*B*c^2*d^4 + 43*B*c^3*d^3 + 40*B* c^4*d^2 - 4*A*c*d^5 - 4*A*c^5*d + 2*B*c*d^5 + 12*B*c^5*d))/(c^2*(c^5 - 3*c ^4*d - 3*c*d^4 + d^5 + 2*c^2*d^3 + 2*c^3*d^2)) + (4*A*c^5 + 3*A*d^5 + 2*B* c^5 - 46*A*c^2*d^3 - 40*A*c^3*d^2 + 28*B*c^2*d^3 + 52*B*c^3*d^2 - 12*A*c*d ^4 - 14*A*c^4*d + 3*B*c*d^4 + 20*B*c^4*d)/(3*(c + d)*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (2*tan(e/2 + (f*x)/2)^3*(6*A*c^6 + 9*A*d^6 + 6*B *c^6 - 177*A*c^2*d^4 - 212*A*c^3*d^3 - 102*A*c^4*d^2 + 105*B*c^2*d^4 + 215 *B*c^3*d^3 + 150*B*c^4*d^2 - 33*A*c*d^5 - 16*A*c^5*d + 9*B*c*d^5 + 40*B*c^ 5*d))/(3*c^2*(c^2 - d^2)*(3*c*d^2 - 3*c^2*d + c^3 - d^3)) + (tan(e/2 + (f* x)/2)*(6*A*c^5 + 6*A*d^5 + 6*B*c^5 - 160*A*c^2*d^3 - 114*A*c^3*d^2 + 97...