Integrand size = 26, antiderivative size = 135 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {35 c^4 x}{2 a^2}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )} \]
35/2*c^4*x/a^2+35/2*c^4*cos(f*x+e)/a^2/f-2/3*a^3*c^4*cos(f*x+e)^7/f/(a+a*s in(f*x+e))^5+14/3*a^4*c^4*cos(f*x+e)^5/f/(a^2+a^2*sin(f*x+e))^3+35/6*c^4*c os(f*x+e)^3/f/(a^2+a^2*sin(f*x+e))
Time = 11.89 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.80 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^4 \left (128 \sin \left (\frac {1}{2} (e+f x)\right )-64 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-640 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+210 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+72 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^2} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(128*Sin[(e + f*x)/2] - 64*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 640*Sin[(e + f*x)/2 ]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 210*(e + f*x)*(Cos[(e + f*x)/2 ] + Sin[(e + f*x)/2])^3 + 72*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x )/2])^3 - 3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12 *f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(a + a*Sin[e + f*x])^2)
Time = 0.65 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 3215, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^4 c^4 \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^4}dx}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^4}dx}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \int \frac {\cos ^4(e+f x)}{(\sin (e+f x) a+a)^2}dx}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \int \frac {\cos (e+f x)^4}{(\sin (e+f x) a+a)^2}dx}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \int \frac {\cos ^2(e+f x)}{\sin (e+f x) a+a}dx}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \int \frac {\cos (e+f x)^2}{\sin (e+f x) a+a}dx}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \left (\frac {3 \left (\frac {\int 1dx}{a}+\frac {\cos (e+f x)}{a f}\right )}{2 a}+\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^4 c^4 \left (-\frac {7 \left (-\frac {5 \left (\frac {\cos ^3(e+f x)}{2 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac {3 \left (\frac {\cos (e+f x)}{a f}+\frac {x}{a}\right )}{2 a}\right )}{a^2}-\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{3 a^2}-\frac {2 \cos ^7(e+f x)}{3 a f (a \sin (e+f x)+a)^5}\right )\) |
a^4*c^4*((-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))
3.3.70.3.1 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 = 0.86 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(\frac {2 c^{4} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(125\) |
| default | \(\frac {2 c^{4} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(125\) |
| risch | \(\frac {35 c^{4} x}{2 a^{2}}+\frac {i c^{4} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {3 c^{4} {\mathrm e}^{i \left (f x +e \right )}}{a^{2} f}+\frac {3 c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{a^{2} f}-\frac {i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {96 i c^{4} {\mathrm e}^{i \left (f x +e \right )}+64 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {160 c^{4}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(150\) |
| parallelrisch | \(\frac {c^{4} \left (1260 f x \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+1260 f x \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-420 f x \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+420 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+1443 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+2493 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-63 \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+3 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+191 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+63 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+1503 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}{24 f \,a^{2} \left (3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) | \(198\) |
| norman | \(\frac {\frac {111 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {131 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a}+\frac {500 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {35 c^{4} x}{2 a}+\frac {164 c^{4}}{3 f a}+\frac {105 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {455 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {315 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {315 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {455 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {105 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {35 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {33 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {460 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {718 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {632 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}+\frac {1454 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}+\frac {815 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(490\) |
2/f*c^4/a^2*((1/2*tan(1/2*f*x+1/2*e)^3+6*tan(1/2*f*x+1/2*e)^2-1/2*tan(1/2* f*x+1/2*e)+6)/(1+tan(1/2*f*x+1/2*e)^2)^2+35/2*arctan(tan(1/2*f*x+1/2*e))-3 2/3/(tan(1/2*f*x+1/2*e)+1)^3+16/(tan(1/2*f*x+1/2*e)+1)^2+16/(tan(1/2*f*x+1 /2*e)+1))
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.56 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=-\frac {3 \, c^{4} \cos \left (f x + e\right )^{4} - 30 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x - 32 \, c^{4} - {\left (105 \, c^{4} f x - 193 \, c^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (105 \, c^{4} f x + 194 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (3 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x + 33 \, c^{4} \cos \left (f x + e\right )^{2} + 32 \, c^{4} + {\left (105 \, c^{4} f x + 226 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
-1/6*(3*c^4*cos(f*x + e)^4 - 30*c^4*cos(f*x + e)^3 + 210*c^4*f*x - 32*c^4 - (105*c^4*f*x - 193*c^4)*cos(f*x + e)^2 + (105*c^4*f*x + 194*c^4)*cos(f*x + e) + (3*c^4*cos(f*x + e)^3 + 210*c^4*f*x + 33*c^4*cos(f*x + e)^2 + 32*c ^4 + (105*c^4*f*x + 226*c^4)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f *x + e))
Leaf count of result is larger than twice the leaf count of optimal. 2312 vs. \(2 (128) = 256\).
Time = 7.28 (sec) , antiderivative size = 2312, normalized size of antiderivative = 17.13 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]
Piecewise((105*c**4*f*x*tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2* f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 315*c**4*f*x*tan(e/ 2 + f*x/2)**6/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)** 6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a** 2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/ 2 + f*x/2) + 6*a**2*f) + 525*c**4*f*x*tan(e/2 + f*x/2)**5/(6*a**2*f*tan(e/ 2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2) **5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a **2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 735*c **4*f*x*tan(e/2 + f*x/2)**4/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan( e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/ 2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18 *a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 735*c**4*f*x*tan(e/2 + f*x/2)**3/(6 *a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*ta n(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f* x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a **2*f) + 525*c**4*f*x*tan(e/2 + f*x/2)**2/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2...
Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (125) = 250\).
Time = 0.30 (sec) , antiderivative size = 903, normalized size of antiderivative = 6.69 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]
1/3*(c^4*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 98*sin(f*x + e)^4 /(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*sin(f* x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e ) + 1) + 5*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^3/ (cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5*a^2*s in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/( cos(f*x + e) + 1))/a^2) + 16*c^4*((12*sin(f*x + e)/(cos(f*x + e) + 1) + 11 *sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(cos(f*x + e) + 1) ^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/ (cos(f*x + e) + 1) + 4*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4*a^2*sin (f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1) ^4 + a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos (f*x + e) + 1))/a^2) + 12*c^4*((9*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin( f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/( cos(f*x + e) + 1)^3) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 2* c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x ...
Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{2}} + \frac {6 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} + \frac {64 \, {\left (3 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, c^{4}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
1/6*(105*(f*x + e)*c^4/a^2 + 6*(c^4*tan(1/2*f*x + 1/2*e)^3 + 12*c^4*tan(1/ 2*f*x + 1/2*e)^2 - c^4*tan(1/2*f*x + 1/2*e) + 12*c^4)/((tan(1/2*f*x + 1/2* e)^2 + 1)^2*a^2) + 64*(3*c^4*tan(1/2*f*x + 1/2*e)^2 + 9*c^4*tan(1/2*f*x + 1/2*e) + 4*c^4)/(a^2*(tan(1/2*f*x + 1/2*e) + 1)^3))/f
Time = 10.67 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.16 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {35\,c^4\,x}{2\,a^2}-\frac {\frac {35\,c^4\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+786\right )}{6}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+328\right )}{6}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+198\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+666\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+974\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+868\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+1428\right )}{6}\right )}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2} \]
(35*c^4*x)/(2*a^2) - ((35*c^4*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((105*c^4* (e + f*x))/2 - (c^4*(315*e + 315*f*x + 786))/6) - (c^4*(105*e + 105*f*x + 328))/6 + tan(e/2 + (f*x)/2)^6*((105*c^4*(e + f*x))/2 - (c^4*(315*e + 315* f*x + 198))/6) + tan(e/2 + (f*x)/2)^5*((175*c^4*(e + f*x))/2 - (c^4*(525*e + 525*f*x + 666))/6) + tan(e/2 + (f*x)/2)^2*((175*c^4*(e + f*x))/2 - (c^4 *(525*e + 525*f*x + 974))/6) + tan(e/2 + (f*x)/2)^4*((245*c^4*(e + f*x))/2 - (c^4*(735*e + 735*f*x + 868))/6) + tan(e/2 + (f*x)/2)^3*((245*c^4*(e + f*x))/2 - (c^4*(735*e + 735*f*x + 1428))/6))/(a^2*f*(tan(e/2 + (f*x)/2) + 1)^3*(tan(e/2 + (f*x)/2)^2 + 1)^2)