Integrand size = 26, antiderivative size = 161 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )} \]
-63/2*c^5*x/a^3-63/2*c^5*cos(f*x+e)/a^3/f-2/5*a^4*c^5*cos(f*x+e)^9/f/(a+a* sin(f*x+e))^7+6/5*a^2*c^5*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^5-42/5*c^5*cos(f *x+e)^5/f/(a+a*sin(f*x+e))^3-21/2*c^5*cos(f*x+e)^3/f/(a^3+a^3*sin(f*x+e))
Time = 16.79 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.88 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-896 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+448 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2304 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-160 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sin (2 (e+f x))\right )}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^3} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 896*Sin[(e + f*x)/ 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 448*(Cos[(e + f*x)/2] + Sin[( e + f*x)/2])^3 + 2304*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])^4 - 630*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 160*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 5*(Cos[(e + f*x)/2] + Sin[ (e + f*x)/2])^5*Sin[2*(e + f*x)]))/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x) /2])^10*(a + a*Sin[e + f*x])^3)
Time = 0.82 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3215, 3042, 3159, 3042, 3159, 3042, 3159, 3042, 3158, 3042, 3161, 24}
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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^5 c^5 \int \frac {\cos ^{10}(e+f x)}{(\sin (e+f x) a+a)^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \int \frac {\cos (e+f x)^{10}}{(\sin (e+f x) a+a)^8}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^6}dx}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^6}dx}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^4}dx}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^4}dx}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \int \frac {\cos ^4(e+f x)}{(\sin (e+f x) a+a)^2}dx}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \int \frac {\cos (e+f x)^4}{(\sin (e+f x) a+a)^2}dx}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \int \frac {\cos ^2(e+f x)}{\sin (e+f x) a+a}dx}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \int \frac {\cos (e+f x)^2}{\sin (e+f x) a+a}dx}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \left (\frac {\int 1dx}{a}+\frac {\cos (e+f x)}{a f}\right )}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^5 c^5 \left (-\frac {9 \left (-\frac {7 \left (-\frac {5 \left (\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac {3 \left (\frac {\cos (e+f x)}{a f}+\frac {x}{a}\right )}{2 a}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )}{5 a^2}-\frac {2 \cos ^9(e+f x)}{5 a f (a \sin (e+f x)+a)^7}\right )\) |
a^5*c^5*((-2*Cos[e + f*x]^9)/(5*a*f*(a + a*Sin[e + f*x])^7) - (9*((-2*Cos[ e + f*x]^7)/(3*a*f*(a + a*Sin[e + f*x])^5) - (7*((-2*Cos[e + f*x]^5)/(a*f* (a + a*Sin[e + f*x])^3) - (5*((3*(x/a + Cos[e + f*x]/(a*f)))/(2*a) + Cos[e + f*x]^3/(2*f*(a^2 + a^2*Sin[e + f*x]))))/a^2))/(3*a^2)))/(5*a^2))
3.3.79.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 1.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
| default | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
| risch | \(-\frac {63 c^{5} x}{2 a^{3}}-\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {4 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{a^{3} f}-\frac {4 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{a^{3} f}+\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {32 \left (-105 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 i c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+25 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}-65 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+18 c^{5}\right )}{5 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(179\) |
| parallelrisch | \(\frac {\left (\left (-2520 f x -4892\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (1260 f x +682\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (252 f x +\frac {5038}{5}\right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (-2520 f x -3044\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-1260 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+252 f x \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )-3286 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-214 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+27 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-\sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )+27 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right ) c^{5}}{8 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(236\) |
| norman | \(\frac {-\frac {945 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}-\frac {431 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a}-\frac {63 c^{5} x}{2 a}-\frac {9765 c^{5} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {4095 c^{5} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2516 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6071 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {7915 c^{5} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {7736 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {5477 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {309 c^{5} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {496 c^{5}}{5 f a}-\frac {4095 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {945 c^{5} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {956 c^{5} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {1179 c^{5} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {4412 c^{5} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2036 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {65 c^{5} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6363 c^{5} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {7640 c^{5} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {19741 c^{5} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {9765 c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(658\) |
2/f*c^5/a^3*(-(1/2*tan(1/2*f*x+1/2*e)^3+8*tan(1/2*f*x+1/2*e)^2-1/2*tan(1/2 *f*x+1/2*e)+8)/(1+tan(1/2*f*x+1/2*e)^2)^2-63/2*arctan(tan(1/2*f*x+1/2*e))- 128/5/(tan(1/2*f*x+1/2*e)+1)^5+64/(tan(1/2*f*x+1/2*e)+1)^4-32/(tan(1/2*f*x +1/2*e)+1)^3-16/(tan(1/2*f*x+1/2*e)+1)^2-32/(tan(1/2*f*x+1/2*e)+1))
Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.77 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {5 \, c^{5} \cos \left (f x + e\right )^{5} + 70 \, c^{5} \cos \left (f x + e\right )^{4} - 1260 \, c^{5} f x - 64 \, c^{5} + 7 \, {\left (45 \, c^{5} f x + 113 \, c^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (945 \, c^{5} f x - 502 \, c^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (315 \, c^{5} f x + 646 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (5 \, c^{5} \cos \left (f x + e\right )^{4} - 65 \, c^{5} \cos \left (f x + e\right )^{3} + 1260 \, c^{5} f x - 64 \, c^{5} - 3 \, {\left (105 \, c^{5} f x - 242 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (315 \, c^{5} f x + 614 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{10 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
-1/10*(5*c^5*cos(f*x + e)^5 + 70*c^5*cos(f*x + e)^4 - 1260*c^5*f*x - 64*c^ 5 + 7*(45*c^5*f*x + 113*c^5)*cos(f*x + e)^3 + (945*c^5*f*x - 502*c^5)*cos( f*x + e)^2 - 2*(315*c^5*f*x + 646*c^5)*cos(f*x + e) - (5*c^5*cos(f*x + e)^ 4 - 65*c^5*cos(f*x + e)^3 + 1260*c^5*f*x - 64*c^5 - 3*(105*c^5*f*x - 242*c ^5)*cos(f*x + e)^2 + 2*(315*c^5*f*x + 614*c^5)*cos(f*x + e))*sin(f*x + e)) /(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4 *a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 3643 vs. \(2 (153) = 306\).
Time = 23.34 (sec) , antiderivative size = 3643, normalized size of antiderivative = 22.63 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
Piecewise((-315*c**5*f*x*tan(e/2 + f*x/2)**9/(10*a**3*f*tan(e/2 + f*x/2)** 9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a **3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*ta n(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 1575*c**5*f*x*tan(e/ 2 + f*x/2)**8/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)* *8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260 *a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f* tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 3780*c**5*f*x*tan(e/2 + f*x/2)**7/(10*a**3*f*tan(e/ 2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2 )**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 2 60*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3* f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 6300*c** 5*f*x*tan(e/2 + f*x/2)**6/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e /2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x /2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3 *f*tan(e/2 + f*x/2) + 10*a**3*f) - 8190*c**5*f*x*tan(e/2 + f*x/2)**5/(10*a **3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*...
Leaf count of result is larger than twice the leaf count of optimal. 1496 vs. \(2 (149) = 298\).
Time = 0.32 (sec) , antiderivative size = 1496, normalized size of antiderivative = 9.29 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
-1/15*(c^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(c os(f*x + e) + 1)^2 + 3805*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f *x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*s in(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^ 2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(co s(f*x + e) + 1)^4 + 26*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*si n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos (f*x + e) + 1)^9) + 195*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 30* c^5*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 + 75*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/ (cos(f*x + e) + 1)^3 + 15*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 11*a^3 *sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e) /(cos(f*x + e) + 1))/a^3) + 20*c^5*((95*sin(f*x + e)/(cos(f*x + e) + 1)...
Time = 0.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {10 \, {\left (c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 16 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 16 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} + \frac {64 \, {\left (10 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 85 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 55 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, c^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{10 \, f} \]
-1/10*(315*(f*x + e)*c^5/a^3 + 10*(c^5*tan(1/2*f*x + 1/2*e)^3 + 16*c^5*tan (1/2*f*x + 1/2*e)^2 - c^5*tan(1/2*f*x + 1/2*e) + 16*c^5)/((tan(1/2*f*x + 1 /2*e)^2 + 1)^2*a^3) + 64*(10*c^5*tan(1/2*f*x + 1/2*e)^4 + 45*c^5*tan(1/2*f *x + 1/2*e)^3 + 85*c^5*tan(1/2*f*x + 1/2*e)^2 + 55*c^5*tan(1/2*f*x + 1/2*e ) + 13*c^5)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f
Time = 11.55 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.26 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {\frac {63\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+4310\right )}{10}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+992\right )}{10}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+650\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+3090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+8814\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+7610\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+12230\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+11090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+14702\right )}{10}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2}-\frac {63\,c^5\,x}{2\,a^3} \]
((63*c^5*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((315*c^5*(e + f*x))/2 - (c^5*( 1575*e + 1575*f*x + 4310))/10) - (c^5*(315*e + 315*f*x + 992))/10 + tan(e/ 2 + (f*x)/2)^8*((315*c^5*(e + f*x))/2 - (c^5*(1575*e + 1575*f*x + 650))/10 ) + tan(e/2 + (f*x)/2)^7*(378*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 30 90))/10) + tan(e/2 + (f*x)/2)^2*(378*c^5*(e + f*x) - (c^5*(3780*e + 3780*f *x + 8814))/10) + tan(e/2 + (f*x)/2)^6*(630*c^5*(e + f*x) - (c^5*(6300*e + 6300*f*x + 7610))/10) + tan(e/2 + (f*x)/2)^3*(630*c^5*(e + f*x) - (c^5*(6 300*e + 6300*f*x + 12230))/10) + tan(e/2 + (f*x)/2)^5*(819*c^5*(e + f*x) - (c^5*(8190*e + 8190*f*x + 11090))/10) + tan(e/2 + (f*x)/2)^4*(819*c^5*(e + f*x) - (c^5*(8190*e + 8190*f*x + 14702))/10))/(a^3*f*(tan(e/2 + (f*x)/2) + 1)^5*(tan(e/2 + (f*x)/2)^2 + 1)^2) - (63*c^5*x)/(2*a^3)