Integrand size = 29, antiderivative size = 89 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {a \cos ^7(e+f x)}{18 f (a+a \sin (e+f x))^8}+\frac {25 \cos ^5(e+f x)}{126 a f (a+a \sin (e+f x))^6}-\frac {47 \cos ^5(e+f x)}{315 a^2 f (a+a \sin (e+f x))^5} \] Output:
-1/18*a*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^8+25/126*cos(f*x+e)^5/a/f/(a+a*sin (f*x+e))^6-47/315*cos(f*x+e)^5/a^2/f/(a+a*sin(f*x+e))^5
Leaf count is larger than twice the leaf count of optimal. \(293\) vs. \(2(89)=178\).
Time = 3.35 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=\frac {1890 \cos \left (\frac {f x}{2}\right )+718830 \cos \left (e+\frac {f x}{2}\right )-467208 \cos \left (e+\frac {3 f x}{2}\right )-1260 \cos \left (2 e+\frac {3 f x}{2}\right )-540 \cos \left (2 e+\frac {5 f x}{2}\right )-179640 \cos \left (3 e+\frac {5 f x}{2}\right )+30753 \cos \left (3 e+\frac {7 f x}{2}\right )+135 \cos \left (4 e+\frac {7 f x}{2}\right )+15 \cos \left (4 e+\frac {9 f x}{2}\right )-15 \cos \left (5 e+\frac {9 f x}{2}\right )+971082 \sin \left (\frac {f x}{2}\right )+1890 \sin \left (e+\frac {f x}{2}\right )+1260 \sin \left (e+\frac {3 f x}{2}\right )+659400 \sin \left (2 e+\frac {3 f x}{2}\right )-303192 \sin \left (2 e+\frac {5 f x}{2}\right )-540 \sin \left (3 e+\frac {5 f x}{2}\right )-135 \sin \left (3 e+\frac {7 f x}{2}\right )-89955 \sin \left (4 e+\frac {7 f x}{2}\right )+13427 \sin \left (4 e+\frac {9 f x}{2}\right )+15 \sin \left (5 e+\frac {9 f x}{2}\right )}{720720 a^7 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \] Input:
Integrate[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]
Output:
(1890*Cos[(f*x)/2] + 718830*Cos[e + (f*x)/2] - 467208*Cos[e + (3*f*x)/2] - 1260*Cos[2*e + (3*f*x)/2] - 540*Cos[2*e + (5*f*x)/2] - 179640*Cos[3*e + ( 5*f*x)/2] + 30753*Cos[3*e + (7*f*x)/2] + 135*Cos[4*e + (7*f*x)/2] + 15*Cos [4*e + (9*f*x)/2] - 15*Cos[5*e + (9*f*x)/2] + 971082*Sin[(f*x)/2] + 1890*S in[e + (f*x)/2] + 1260*Sin[e + (3*f*x)/2] + 659400*Sin[2*e + (3*f*x)/2] - 303192*Sin[2*e + (5*f*x)/2] - 540*Sin[3*e + (5*f*x)/2] - 135*Sin[3*e + (7* f*x)/2] - 89955*Sin[4*e + (7*f*x)/2] + 13427*Sin[4*e + (9*f*x)/2] + 15*Sin [5*e + (9*f*x)/2])/(720720*a^7*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^9)
Time = 0.71 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3351, 2009}
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.
| \(\displaystyle \int \frac {\sin ^2(e+f x) \cos ^4(e+f x)}{(a \sin (e+f x)+a)^7} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x)^2 \cos (e+f x)^4}{(a \sin (e+f x)+a)^7}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \sec ^8(e+f x) (a-a \sin (e+f x))^7 \tan ^2(e+f x)dx}{a^{14}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (e+f x)^2 (a-a \sin (e+f x))^7}{\cos (e+f x)^{10}}dx}{a^{14}}\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\frac {1}{a^3 (\sin (e+f x)+1)}-\frac {6}{a^3 (\sin (e+f x)+1)^2}+\frac {13}{a^3 (\sin (e+f x)+1)^3}-\frac {12}{a^3 (\sin (e+f x)+1)^4}+\frac {4}{a^3 (\sin (e+f x)+1)^5}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {47 \cos (e+f x)}{315 a^3 f (\sin (e+f x)+1)}+\frac {268 \cos (e+f x)}{315 a^3 f (\sin (e+f x)+1)^2}-\frac {181 \cos (e+f x)}{105 a^3 f (\sin (e+f x)+1)^3}+\frac {92 \cos (e+f x)}{63 a^3 f (\sin (e+f x)+1)^4}-\frac {4 \cos (e+f x)}{9 a^3 f (\sin (e+f x)+1)^5}}{a^4}\) |
Input:
Int[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]
Output:
((-4*Cos[e + f*x])/(9*a^3*f*(1 + Sin[e + f*x])^5) + (92*Cos[e + f*x])/(63* a^3*f*(1 + Sin[e + f*x])^4) - (181*Cos[e + f*x])/(105*a^3*f*(1 + Sin[e + f *x])^3) + (268*Cos[e + f*x])/(315*a^3*f*(1 + Sin[e + f*x])^2) - (47*Cos[e + f*x])/(315*a^3*f*(1 + Sin[e + f*x])))/a^4
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 0.92 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.12
| method | result | size |
| parallelrisch | \(\frac {-\frac {4}{315}-\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {16 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{5}-\frac {28 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}}{f \,a^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}\) | \(100\) |
| derivativedivides | \(\frac {-\frac {328}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {832}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {8}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {352}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{7}}\) | \(115\) |
| default | \(\frac {-\frac {328}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {832}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {8}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {352}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{7}}\) | \(115\) |
| risch | \(-\frac {2 \left (-2520 i {\mathrm e}^{5 i \left (f x +e \right )}-2310 \,{\mathrm e}^{6 i \left (f x +e \right )}+3402 \,{\mathrm e}^{4 i \left (f x +e \right )}+630 i {\mathrm e}^{7 i \left (f x +e \right )}+315 \,{\mathrm e}^{8 i \left (f x +e \right )}+1638 i {\mathrm e}^{3 i \left (f x +e \right )}-1062 \,{\mathrm e}^{2 i \left (f x +e \right )}-108 i {\mathrm e}^{i \left (f x +e \right )}+47\right )}{315 f \,a^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{9}}\) | \(117\) |
Input:
int(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x,method=_RETURNVERBOSE)
Output:
4/315*(-1-9*tan(1/2*f*x+1/2*e)-36*tan(1/2*f*x+1/2*e)^2+126*tan(1/2*f*x+1/2 *e)^3-441*tan(1/2*f*x+1/2*e)^4+315*tan(1/2*f*x+1/2*e)^5-210*tan(1/2*f*x+1/ 2*e)^6)/f/a^7/(tan(1/2*f*x+1/2*e)+1)^9
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (83) = 166\).
Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.73 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} - 115 \, \cos \left (f x + e\right )^{3} - 265 \, \cos \left (f x + e\right )^{2} - {\left (47 \, \cos \left (f x + e\right )^{4} - 80 \, \cos \left (f x + e\right )^{3} - 195 \, \cos \left (f x + e\right )^{2} + 70 \, \cos \left (f x + e\right ) + 140\right )} \sin \left (f x + e\right ) + 70 \, \cos \left (f x + e\right ) + 140}{315 \, {\left (a^{7} f \cos \left (f x + e\right )^{5} + 5 \, a^{7} f \cos \left (f x + e\right )^{4} - 8 \, a^{7} f \cos \left (f x + e\right )^{3} - 20 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f + {\left (a^{7} f \cos \left (f x + e\right )^{4} - 4 \, a^{7} f \cos \left (f x + e\right )^{3} - 12 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="frica s")
Output:
-1/315*(47*cos(f*x + e)^5 + 127*cos(f*x + e)^4 - 115*cos(f*x + e)^3 - 265* cos(f*x + e)^2 - (47*cos(f*x + e)^4 - 80*cos(f*x + e)^3 - 195*cos(f*x + e) ^2 + 70*cos(f*x + e) + 140)*sin(f*x + e) + 70*cos(f*x + e) + 140)/(a^7*f*c os(f*x + e)^5 + 5*a^7*f*cos(f*x + e)^4 - 8*a^7*f*cos(f*x + e)^3 - 20*a^7*f *cos(f*x + e)^2 + 8*a^7*f*cos(f*x + e) + 16*a^7*f + (a^7*f*cos(f*x + e)^4 - 4*a^7*f*cos(f*x + e)^3 - 12*a^7*f*cos(f*x + e)^2 + 8*a^7*f*cos(f*x + e) + 16*a^7*f)*sin(f*x + e))
Timed out. \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**4*sin(f*x+e)**2/(a+a*sin(f*x+e))**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (83) = 166\).
Time = 0.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.76 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {126 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {315 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {210 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (a^{7} + \frac {9 \, a^{7} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{7} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{7} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \] Input:
integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="maxim a")
Output:
-4/315*(9*sin(f*x + e)/(cos(f*x + e) + 1) + 36*sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 - 126*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 441*sin(f*x + e)^4/(c os(f*x + e) + 1)^4 - 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 210*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1)/((a^7 + 9*a^7*sin(f*x + e)/(cos(f*x + e) + 1) + 36*a^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 84*a^7*sin(f*x + e)^3 /(cos(f*x + e) + 1)^3 + 126*a^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 126* a^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*a^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 36*a^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*a^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*f)
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4 \, {\left (210 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 126 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{315 \, a^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \] Input:
integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="giac" )
Output:
-4/315*(210*tan(1/2*f*x + 1/2*e)^6 - 315*tan(1/2*f*x + 1/2*e)^5 + 441*tan( 1/2*f*x + 1/2*e)^4 - 126*tan(1/2*f*x + 1/2*e)^3 + 36*tan(1/2*f*x + 1/2*e)^ 2 + 9*tan(1/2*f*x + 1/2*e) + 1)/(a^7*f*(tan(1/2*f*x + 1/2*e) + 1)^9)
Time = 17.87 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=-\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+9\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-126\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+210\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{315\,a^7\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \] Input:
int((cos(e + f*x)^4*sin(e + f*x)^2)/(a + a*sin(e + f*x))^7,x)
Output:
-(4*cos(e/2 + (f*x)/2)^3*(cos(e/2 + (f*x)/2)^6 + 210*sin(e/2 + (f*x)/2)^6 - 315*cos(e/2 + (f*x)/2)*sin(e/2 + (f*x)/2)^5 + 9*cos(e/2 + (f*x)/2)^5*sin (e/2 + (f*x)/2) + 441*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^4 - 126*cos( e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^3 + 36*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^2))/(315*a^7*f*(cos(e/2 + (f*x)/2) + sin(e/2 + (f*x)/2))^9)
Time = 0.17 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.28 \[ \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx=\frac {-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-\frac {28 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{5}-\frac {16 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35}-\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {4}{315}}{a^{7} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \] Input:
int(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x)
Output:
(4*( - 210*tan((e + f*x)/2)**6 + 315*tan((e + f*x)/2)**5 - 441*tan((e + f* x)/2)**4 + 126*tan((e + f*x)/2)**3 - 36*tan((e + f*x)/2)**2 - 9*tan((e + f *x)/2) - 1))/(315*a**7*f*(tan((e + f*x)/2)**9 + 9*tan((e + f*x)/2)**8 + 36 *tan((e + f*x)/2)**7 + 84*tan((e + f*x)/2)**6 + 126*tan((e + f*x)/2)**5 + 126*tan((e + f*x)/2)**4 + 84*tan((e + f*x)/2)**3 + 36*tan((e + f*x)/2)**2 + 9*tan((e + f*x)/2) + 1))