Integrand size = 25, antiderivative size = 364 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=-\frac {d^3 \left (20 c^2+30 c d+13 d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{27 (c-d)^5 (c+d)^2 \sqrt {c^2-d^2} f}-\frac {d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{810 (c-d)^4 (c+d) f (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{5 (c-d) f (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^2}-\frac {(2 c-11 d) \cos (e+f x)}{45 (c-d)^2 f (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{15 (c-d)^3 f (27+27 \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{810 (c-d)^5 (c+d)^2 f (c+d \sin (e+f x))} \]
-1/30*d*(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*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+ e))^2-1/15*(2*c-11*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*(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*(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))-d^3*(20*c^2+30*c*d+13 *d^2)*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 /f/(c^2-d^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(911\) vs. \(2(364)=728\).
Time = 7.92 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(3+3 \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 (-\frac {480 d^3 \left (20 c^2+30 c d+13 d^2\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}{\sqrt {c^2-d^2}}+\frac {10 d \left (-40 c^5+340 c^4 d+1934 c^3 d^2+3040 c^2 d^3+1994 c d^4+481 d^5\right ) \cos \left (\frac {1}{2} (e+f x)\right )-2 \left (80 c^6-424 c^5 d+1200 c^4 d^2+9698 c^3 d^3+17640 c^2 d^4+12371 c d^5+2905 d^6\right ) \cos \left (\frac {3}{2} (e+f x)\right )-1260 c^3 d^3 \cos \left (\frac {5}{2} (e+f x)\right )-2640 c^2 d^4 \cos \left (\frac {5}{2} (e+f x)\right )-2250 c d^5 \cos \left (\frac {5}{2} (e+f x)\right )-870 d^6 \cos \left (\frac {5}{2} (e+f x)\right )+32 c^5 d \cos \left (\frac {7}{2} (e+f x)\right )-200 c^4 d^2 \cos \left (\frac {7}{2} (e+f x)\right )+836 c^3 d^3 \cos \left (\frac {7}{2} (e+f x)\right )+4480 c^2 d^4 \cos \left (\frac {7}{2} (e+f x)\right )+5747 c d^5 \cos \left (\frac {7}{2} (e+f x)\right )+2200 d^6 \cos \left (\frac {7}{2} (e+f x)\right )-135 c d^5 \cos \left (\frac {9}{2} (e+f x)\right )-90 d^6 \cos \left (\frac {9}{2} (e+f x)\right )+320 c^6 \sin \left (\frac {1}{2} (e+f x)\right )-1520 c^5 d \sin \left (\frac {1}{2} (e+f x)\right )+4568 c^4 d^2 \sin \left (\frac {1}{2} (e+f x)\right )+27340 c^3 d^3 \sin \left (\frac {1}{2} (e+f x)\right )+40904 c^2 d^4 \sin \left (\frac {1}{2} (e+f x)\right )+26020 c d^5 \sin \left (\frac {1}{2} (e+f x)\right )+6318 d^6 \sin \left (\frac {1}{2} (e+f x)\right )+800 c^4 d^2 \sin \left (\frac {3}{2} (e+f x)\right )+7500 c^3 d^3 \sin \left (\frac {3}{2} (e+f x)\right )+13280 c^2 d^4 \sin \left (\frac {3}{2} (e+f x)\right )+9690 c d^5 \sin \left (\frac {3}{2} (e+f x)\right )+2750 d^6 \sin \left (\frac {3}{2} (e+f x)\right )-32 c^6 \sin \left (\frac {5}{2} (e+f x)\right )+80 c^5 d \sin \left (\frac {5}{2} (e+f x)\right )-32 c^4 d^2 \sin \left (\frac {5}{2} (e+f x)\right )-6820 c^3 d^3 \sin \left (\frac {5}{2} (e+f x)\right )-18080 c^2 d^4 \sin \left (\frac {5}{2} (e+f x)\right )-15670 c d^5 \sin \left (\frac {5}{2} (e+f x)\right )-4266 d^6 \sin \left (\frac {5}{2} (e+f x)\right )-60 c^2 d^4 \sin \left (\frac {7}{2} (e+f x)\right )+135 c d^5 \sin \left (\frac {7}{2} (e+f x)\right )+60 d^6 \sin \left (\frac {7}{2} (e+f x)\right )+8 c^4 d^2 \sin \left (\frac {9}{2} (e+f x)\right )-60 c^3 d^3 \sin \left (\frac {9}{2} (e+f x)\right )+284 c^2 d^4 \sin \left (\frac {9}{2} (e+f x)\right )+915 c d^5 \sin \left (\frac {9}{2} (e+f x)\right )+518 d^6 \sin \left (\frac {9}{2} (e+f x)\right )}{(c+d \sin (e+f x))^2}\right )}{12960 (c-d)^5 (c+d)^2 f (1+\sin (e+f x))^3} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((-480*d^3*(20*c^2 + 30*c*d + 13*d^ 2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Si n[(e + f*x)/2])^5)/Sqrt[c^2 - d^2] + (10*d*(-40*c^5 + 340*c^4*d + 1934*c^3 *d^2 + 3040*c^2*d^3 + 1994*c*d^4 + 481*d^5)*Cos[(e + f*x)/2] - 2*(80*c^6 - 424*c^5*d + 1200*c^4*d^2 + 9698*c^3*d^3 + 17640*c^2*d^4 + 12371*c*d^5 + 2 905*d^6)*Cos[(3*(e + f*x))/2] - 1260*c^3*d^3*Cos[(5*(e + f*x))/2] - 2640*c ^2*d^4*Cos[(5*(e + f*x))/2] - 2250*c*d^5*Cos[(5*(e + f*x))/2] - 870*d^6*Co s[(5*(e + f*x))/2] + 32*c^5*d*Cos[(7*(e + f*x))/2] - 200*c^4*d^2*Cos[(7*(e + f*x))/2] + 836*c^3*d^3*Cos[(7*(e + f*x))/2] + 4480*c^2*d^4*Cos[(7*(e + f*x))/2] + 5747*c*d^5*Cos[(7*(e + f*x))/2] + 2200*d^6*Cos[(7*(e + f*x))/2] - 135*c*d^5*Cos[(9*(e + f*x))/2] - 90*d^6*Cos[(9*(e + f*x))/2] + 320*c^6* Sin[(e + f*x)/2] - 1520*c^5*d*Sin[(e + f*x)/2] + 4568*c^4*d^2*Sin[(e + f*x )/2] + 27340*c^3*d^3*Sin[(e + f*x)/2] + 40904*c^2*d^4*Sin[(e + f*x)/2] + 2 6020*c*d^5*Sin[(e + f*x)/2] + 6318*d^6*Sin[(e + f*x)/2] + 800*c^4*d^2*Sin[ (3*(e + f*x))/2] + 7500*c^3*d^3*Sin[(3*(e + f*x))/2] + 13280*c^2*d^4*Sin[( 3*(e + f*x))/2] + 9690*c*d^5*Sin[(3*(e + f*x))/2] + 2750*d^6*Sin[(3*(e + f *x))/2] - 32*c^6*Sin[(5*(e + f*x))/2] + 80*c^5*d*Sin[(5*(e + f*x))/2] - 32 *c^4*d^2*Sin[(5*(e + f*x))/2] - 6820*c^3*d^3*Sin[(5*(e + f*x))/2] - 18080* c^2*d^4*Sin[(5*(e + f*x))/2] - 15670*c*d^5*Sin[(5*(e + f*x))/2] - 4266*d^6 *Sin[(5*(e + f*x))/2] - 60*c^2*d^4*Sin[(7*(e + f*x))/2] + 135*c*d^5*Sin...
Time = 1.75 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 3245, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {a (2 c-7 d)+4 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (2 c-7 d)+4 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (2 c-7 d)+4 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^3}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (2 c^2-9 d c+43 d^2\right ) a^2+3 (2 c-11 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (2 c^2-9 d c+43 d^2\right ) a^2+3 (2 c-11 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (2 c^2-9 d c+43 d^2\right ) a^2+3 (2 c-11 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^3}dx}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 (2 c-65 d) d^2 a^3+2 d \left (2 c^2-15 d c+76 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 (2 c-65 d) d^2 a^3+2 d \left (2 c^2-15 d c+76 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 (2 c-65 d) d^2 a^3+2 d \left (2 c^2-15 d c+76 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^3}dx}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {2 d^2 \left (2 c^2-165 d c-152 d^2\right ) a^3+d \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 d^2 \left (2 c^2-165 d c-152 d^2\right ) a^3+d \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 d^2 \left (2 c^2-165 d c-152 d^2\right ) a^3+d \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int \frac {15 a^3 d^3 \left (20 c^2+30 d c+13 d^2\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {15 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {30 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {60 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {\frac {-\frac {30 a^3 d^3 \left (20 c^2+30 c d+13 d^2\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {a^3 d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^3 d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{a^2 (c-d)}-\frac {a^2 \left (2 c^2-15 c d+76 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}}{3 a^2 (c-d)}-\frac {a (2 c-11 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}\) |
-1/5*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*c - 11*d)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^2* (c + d*Sin[e + f*x])^2) + (-((a^2*(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*(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^3*(20*c^2 + 30*c*d + 13*d^2)*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*(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))
3.5.79.3.1 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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*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*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
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 = 5.56 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {-4 c +10 d}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c -14 d \right )}{3 \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (c^{2}-5 c d +10 d^{2}\right )}{\left (c -d \right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{5 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 d^{3} \left (\frac {\frac {d^{2} \left (11 c^{2}+6 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 c^{4}+6 c^{3} d +19 c^{2} d^{2}+12 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (29 c^{2}+18 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (10 c^{2}+6 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (20 c^{2}+30 c d +13 d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{5}}}{a^{3} f}\) | \(446\) |
| default | \(\frac {-\frac {-4 c +10 d}{\left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c -14 d \right )}{3 \left (c -d \right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (c^{2}-5 c d +10 d^{2}\right )}{\left (c -d \right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{5 \left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {4}{\left (c -d \right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 d^{3} \left (\frac {\frac {d^{2} \left (11 c^{2}+6 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (10 c^{4}+6 c^{3} d +19 c^{2} d^{2}+12 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{2}+2 c d +d^{2}\right )}+\frac {d^{2} \left (29 c^{2}+18 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}+\frac {d \left (10 c^{2}+6 c d -d^{2}\right )}{2 c^{2}+4 c d +2 d^{2}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (20 c^{2}+30 c d +13 d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right )^{5}}}{a^{3} f}\) | \(446\) |
| risch | \(\text {Expression too large to display}\) | \(1345\) |
2/f/a^3*(-1/2*(-4*c+10*d)/(c-d)^4/(tan(1/2*f*x+1/2*e)+1)^2-1/3*(8*c-14*d)/ (c-d)^4/(tan(1/2*f*x+1/2*e)+1)^3-(c^2-5*c*d+10*d^2)/(c-d)^5/(tan(1/2*f*x+1 /2*e)+1)-4/5/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)^5+2/(c-d)^3/(tan(1/2*f*x+1/2*e )+1)^4-d^3/(c-d)^5*((1/2*d^2*(11*c^2+6*c*d-2*d^2)/c/(c^2+2*c*d+d^2)*tan(1/ 2*f*x+1/2*e)^3+1/2*d*(10*c^4+6*c^3*d+19*c^2*d^2+12*c*d^3-2*d^4)/c^2/(c^2+2 *c*d+d^2)*tan(1/2*f*x+1/2*e)^2+1/2*d^2*(29*c^2+18*c*d-2*d^2)/(c^2+2*c*d+d^ 2)/c*tan(1/2*f*x+1/2*e)+1/2*d*(10*c^2+6*c*d-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*c^2+30*c*d+13*d^2)/(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. 2571 vs. \(2 (363) = 726\).
Time = 0.47 (sec) , antiderivative size = 5226, normalized size of antiderivative = 14.36 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 766 vs. \(2 (363) = 726\).
Time = 0.56 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=-\frac {\frac {15 \, {\left (20 \, c^{2} d^{3} + 30 \, c d^{4} + 13 \, d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (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}\right )} \sqrt {c^{2} - d^{2}}} + \frac {15 \, {\left (11 \, c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 10 \, c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 19 \, c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 29 \, c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 10 \, c^{4} d^{4} + 6 \, c^{3} d^{5} - c^{2} d^{6}\right )}}{{\left (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}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}} + \frac {2 \, {\left (15 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 75 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 150 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 195 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 525 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 245 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 745 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 145 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 485 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{2} - 44 \, c d + 127 \, d^{2}\right )}}{{\left (a^{3} c^{5} - 5 \, a^{3} c^{4} d + 10 \, a^{3} c^{3} d^{2} - 10 \, a^{3} c^{2} d^{3} + 5 \, a^{3} c d^{4} - a^{3} d^{5}\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \]
-1/15*(15*(20*c^2*d^3 + 30*c*d^4 + 13*d^5)*(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^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*(11*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 6*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 - 2*c*d^7*tan(1/2*f*x + 1/2*e )^3 + 10*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 6*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 19*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 12*c*d^7*tan(1/2*f*x + 1/2*e)^2 - 2 *d^8*tan(1/2*f*x + 1/2*e)^2 + 29*c^3*d^5*tan(1/2*f*x + 1/2*e) + 18*c^2*d^6 *tan(1/2*f*x + 1/2*e) - 2*c*d^7*tan(1/2*f*x + 1/2*e) + 10*c^4*d^4 + 6*c^3* d^5 - 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*c^2*tan(1/2*f*x + 1/2*e )^4 - 75*c*d*tan(1/2*f*x + 1/2*e)^4 + 150*d^2*tan(1/2*f*x + 1/2*e)^4 + 30* c^2*tan(1/2*f*x + 1/2*e)^3 - 195*c*d*tan(1/2*f*x + 1/2*e)^3 + 525*d^2*tan( 1/2*f*x + 1/2*e)^3 + 40*c^2*tan(1/2*f*x + 1/2*e)^2 - 245*c*d*tan(1/2*f*x + 1/2*e)^2 + 745*d^2*tan(1/2*f*x + 1/2*e)^2 + 20*c^2*tan(1/2*f*x + 1/2*e) - 145*c*d*tan(1/2*f*x + 1/2*e) + 485*d^2*tan(1/2*f*x + 1/2*e) + 7*c^2 - 44* c*d + 127*d^2)/((a^3*c^5 - 5*a^3*c^4*d + 10*a^3*c^3*d^2 - 10*a^3*c^2*d^3 + 5*a^3*c*d^4 - a^3*d^5)*(tan(1/2*f*x + 1/2*e) + 1)^5))/f
Time = 12.00 (sec) , antiderivative size = 1660, normalized size of antiderivative = 4.56 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
(d^3*atan(((d^3*(30*c*d + 20*c^2 + 13*d^2)*(2*a^3*d^8 - 6*a^3*c*d^7 - 2*a^ 3*c^7*d + 2*a^3*c^2*d^6 + 10*a^3*c^3*d^5 - 10*a^3*c^4*d^4 - 2*a^3*c^5*d^3 + 6*a^3*c^6*d^2))/(2*a^3*(c + d)^(5/2)*(c - d)^(11/2)) - (c*d^3*tan(e/2 + (f*x)/2)*(30*c*d + 20*c^2 + 13*d^2)*(a^3*c^7 - a^3*d^7 + 3*a^3*c*d^6 - 3*a ^3*c^6*d - a^3*c^2*d^5 - 5*a^3*c^3*d^4 + 5*a^3*c^4*d^3 + a^3*c^5*d^2))/(a^ 3*(c + d)^(5/2)*(c - d)^(11/2)))/(30*c*d^4 + 13*d^5 + 20*c^2*d^3))*(30*c*d + 20*c^2 + 13*d^2))/(a^3*f*(c + d)^(5/2)*(c - d)^(11/2)) - ((90*c*d^5 - 6 0*c^5*d + 14*c^6 - 15*d^6 + 404*c^2*d^4 + 420*c^3*d^3 + 92*c^4*d^2)/(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*c*d^7 - 10*c^7*d + 4*c^8 - 2*d^8 + 49*c^2*d^6 + 141*c^3*d^5 + 200*c^4*d^4 + 122*c^5*d^3 - 2*c^6*d^2))/(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*(114 *c*d^7 - 54*c^7*d + 28*c^8 - 30*d^8 + 759*c^2*d^6 + 1707*c^3*d^5 + 1960*c^ 4*d^4 + 870*c^5*d^3 - 62*c^6*d^2))/(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*(270*c*d^ 7 - 62*c^7*d + 32*c^8 - 60*d^8 + 1857*c^2*d^6 + 3763*c^3*d^5 + 3560*c^4*d^ 4 + 1294*c^5*d^3 - 70*c^6*d^2))/(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)^8*(6*c*d^6 - 6 *c^6*d + 2*c^7 - 2*d^7 + 11*c^2*d^5 + 20*c^3*d^4 + 30*c^4*d^3 + 2*c^5*d^2) )/(c*(c - d)*(2*c*d + c^2 + d^2)*(c^4 - 4*c^3*d - 4*c*d^3 + d^4 + 6*c^2...