Integrand size = 27, antiderivative size = 144 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}+\frac {4 \cos (e+f x)}{315 f \left (a^6+a^6 \sin (e+f x)\right )} \] Output:
2/9*cos(f*x+e)/a/f/(a+a*sin(f*x+e))^5-19/63*cos(f*x+e)/a^2/f/(a+a*sin(f*x+ e))^4+2/105*cos(f*x+e)/f/(a^2+a^2*sin(f*x+e))^3+4/315*cos(f*x+e)/f/(a^3+a^ 3*sin(f*x+e))^2+4/315*cos(f*x+e)/f/(a^6+a^6*sin(f*x+e))
Time = 1.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {378 \cos \left (e+\frac {f x}{2}\right )+210 \cos \left (e+\frac {3 f x}{2}\right )-108 \cos \left (3 e+\frac {5 f x}{2}\right )+225 \cos \left (3 e+\frac {7 f x}{2}\right )+3 \cos \left (5 e+\frac {9 f x}{2}\right )+3150 \sin \left (\frac {f x}{2}\right )+2562 \sin \left (2 e+\frac {3 f x}{2}\right )-900 \sin \left (2 e+\frac {5 f x}{2}\right )-27 \sin \left (4 e+\frac {7 f x}{2}\right )+25 \sin \left (4 e+\frac {9 f x}{2}\right )}{13860 a^6 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \] Input:
Integrate[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]
Output:
-1/13860*(378*Cos[e + (f*x)/2] + 210*Cos[e + (3*f*x)/2] - 108*Cos[3*e + (5 *f*x)/2] + 225*Cos[3*e + (7*f*x)/2] + 3*Cos[5*e + (9*f*x)/2] + 3150*Sin[(f *x)/2] + 2562*Sin[2*e + (3*f*x)/2] - 900*Sin[2*e + (5*f*x)/2] - 27*Sin[4*e + (7*f*x)/2] + 25*Sin[4*e + (9*f*x)/2])/(a^6*f*(Cos[e/2] + Sin[e/2])*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^9)
Time = 0.72 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3336, 25, 3042, 3229, 3042, 3129, 3042, 3129, 3042, 3127}
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.
| \(\displaystyle \int \frac {\sin (e+f x) \cos ^2(e+f x)}{(a \sin (e+f x)+a)^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x) \cos (e+f x)^2}{(a \sin (e+f x)+a)^6}dx\) |
| \(\Big \downarrow \) 3336 |
| \(\displaystyle \frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}-\frac {\int -\frac {10 a-9 a \sin (e+f x)}{(\sin (e+f x) a+a)^4}dx}{9 a^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {10 a-9 a \sin (e+f x)}{(\sin (e+f x) a+a)^4}dx}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {10 a-9 a \sin (e+f x)}{(\sin (e+f x) a+a)^4}dx}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle \frac {-\frac {6}{7} \int \frac {1}{(\sin (e+f x) a+a)^3}dx-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {6}{7} \int \frac {1}{(\sin (e+f x) a+a)^3}dx-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {-\frac {6}{7} \left (\frac {2 \int \frac {1}{(\sin (e+f x) a+a)^2}dx}{5 a}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\right )-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {6}{7} \left (\frac {2 \int \frac {1}{(\sin (e+f x) a+a)^2}dx}{5 a}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\right )-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {-\frac {6}{7} \left (\frac {2 \left (\frac {\int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\right )-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {6}{7} \left (\frac {2 \left (\frac {\int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\right )-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {-\frac {19 a \cos (e+f x)}{7 f (a \sin (e+f x)+a)^4}-\frac {6}{7} \left (\frac {2 \left (-\frac {\cos (e+f x)}{3 a f (a \sin (e+f x)+a)}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\right )}{9 a^3}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}\) |
Input:
Int[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]
Output:
(2*Cos[e + f*x])/(9*a*f*(a + a*Sin[e + f*x])^5) + ((-19*a*Cos[e + f*x])/(7 *f*(a + a*Sin[e + f*x])^4) - (6*(-1/5*Cos[e + f*x]/(f*(a + a*Sin[e + f*x]) ^3) + (2*(-1/3*Cos[e + f*x]/(f*(a + a*Sin[e + f*x])^2) - Cos[e + f*x]/(3*a *f*(a + a*Sin[e + f*x]))))/(5*a)))/7)/(9*a^3)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
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/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*( (c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos [e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Simp[1/(b^ 3*(2*m + 3)) Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d* (2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(-\frac {8 \left (105 \,{\mathrm e}^{6 i \left (f x +e \right )}+21 i {\mathrm e}^{3 i \left (f x +e \right )}-126 \,{\mathrm e}^{4 i \left (f x +e \right )}+9 i {\mathrm e}^{i \left (f x +e \right )}+36 \,{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{315 f \,a^{6} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{9}}\) | \(82\) |
| parallelrisch | \(\frac {-\frac {22}{315}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35}-\frac {58 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15}-\frac {14 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{f \,a^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}\) | \(113\) |
| derivativedivides | \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}}{f \,a^{6}}\) | \(130\) |
| default | \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}}{f \,a^{6}}\) | \(130\) |
| norman | \(\frac {-\frac {22}{315 a f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{a f}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{a f}-\frac {18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{a f}-\frac {242 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{315 a f}-\frac {174 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{5 a f}-\frac {274 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{35 a f}-\frac {646 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{315 a f}-\frac {862 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{15 a f}-\frac {1202 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{35 a f}-\frac {1762 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{105 a f}-\frac {8342 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{105 a f}-\frac {9442 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{105 a f}-\frac {17494 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{315 a f}-\frac {23858 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{315 a f}-\frac {28786 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{315 a f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3} a^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}\) | \(325\) |
Input:
int(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
Output:
-8/315*(105*exp(6*I*(f*x+e))+21*I*exp(3*I*(f*x+e))-126*exp(4*I*(f*x+e))+9* I*exp(I*(f*x+e))+36*exp(2*I*(f*x+e))-1)/f/a^6/(exp(I*(f*x+e))+I)^9
Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4 \, \cos \left (f x + e\right )^{5} - 16 \, \cos \left (f x + e\right )^{4} - 50 \, \cos \left (f x + e\right )^{3} - 65 \, \cos \left (f x + e\right )^{2} - {\left (4 \, \cos \left (f x + e\right )^{4} + 20 \, \cos \left (f x + e\right )^{3} - 30 \, \cos \left (f x + e\right )^{2} + 35 \, \cos \left (f x + e\right ) + 70\right )} \sin \left (f x + e\right ) + 35 \, \cos \left (f x + e\right ) + 70}{315 \, {\left (a^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{6} f \cos \left (f x + e\right )^{4} - 8 \, a^{6} f \cos \left (f x + e\right )^{3} - 20 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f + {\left (a^{6} f \cos \left (f x + e\right )^{4} - 4 \, a^{6} f \cos \left (f x + e\right )^{3} - 12 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="fricas" )
Output:
1/315*(4*cos(f*x + e)^5 - 16*cos(f*x + e)^4 - 50*cos(f*x + e)^3 - 65*cos(f *x + e)^2 - (4*cos(f*x + e)^4 + 20*cos(f*x + e)^3 - 30*cos(f*x + e)^2 + 35 *cos(f*x + e) + 70)*sin(f*x + e) + 35*cos(f*x + e) + 70)/(a^6*f*cos(f*x + e)^5 + 5*a^6*f*cos(f*x + e)^4 - 8*a^6*f*cos(f*x + e)^3 - 20*a^6*f*cos(f*x + e)^2 + 8*a^6*f*cos(f*x + e) + 16*a^6*f + (a^6*f*cos(f*x + e)^4 - 4*a^6*f *cos(f*x + e)^3 - 12*a^6*f*cos(f*x + e)^2 + 8*a^6*f*cos(f*x + e) + 16*a^6* f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (131) = 262\).
Time = 47.29 (sec) , antiderivative size = 1501, normalized size of antiderivative = 10.42 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate(cos(f*x+e)**2*sin(f*x+e)/(a+a*sin(f*x+e))**6,x)
Output:
Piecewise((-630*tan(e/2 + f*x/2)**7/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835 *a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a** 6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f* tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan( e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 630*tan(e/2 + f*x/2)**6/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2 )**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 2 6460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835* a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 1890*tan(e/2 + f*x/2)**5/(315*a**6 *f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*ta n(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/ 2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2 ) + 315*a**6*f) - 882*tan(e/2 + f*x/2)**4/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 264 60*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a **6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6* f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 1218* tan(e/2 + f*x/2)**3/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e...
Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (134) = 268\).
Time = 0.04 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (\frac {99 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {81 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {609 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {945 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {315 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 11\right )}}{315 \, {\left (a^{6} + \frac {9 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{6} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{6} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \] Input:
integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="maxima" )
Output:
-2/315*(99*sin(f*x + e)/(cos(f*x + e) + 1) + 81*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 609*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 441*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 + 945*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 315*sin(f* x + e)^6/(cos(f*x + e) + 1)^6 + 315*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 11)/((a^6 + 9*a^6*sin(f*x + e)/(cos(f*x + e) + 1) + 36*a^6*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 + 84*a^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*a^ 6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 126*a^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*a^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 36*a^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*a^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^ 6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*f)
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 945 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 609 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 81 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 99 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11\right )}}{315 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \] Input:
integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="giac")
Output:
-2/315*(315*tan(1/2*f*x + 1/2*e)^7 + 315*tan(1/2*f*x + 1/2*e)^6 + 945*tan( 1/2*f*x + 1/2*e)^5 + 441*tan(1/2*f*x + 1/2*e)^4 + 609*tan(1/2*f*x + 1/2*e) ^3 + 81*tan(1/2*f*x + 1/2*e)^2 + 99*tan(1/2*f*x + 1/2*e) + 11)/(a^6*f*(tan (1/2*f*x + 1/2*e) + 1)^9)
Time = 17.91 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+99\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+81\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+609\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+315\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{315\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \] Input:
int((cos(e + f*x)^2*sin(e + f*x))/(a + a*sin(e + f*x))^6,x)
Output:
-(2*cos(e/2 + (f*x)/2)^2*(11*cos(e/2 + (f*x)/2)^7 + 315*sin(e/2 + (f*x)/2) ^7 + 315*cos(e/2 + (f*x)/2)*sin(e/2 + (f*x)/2)^6 + 99*cos(e/2 + (f*x)/2)^6 *sin(e/2 + (f*x)/2) + 945*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^5 + 441* cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^4 + 609*cos(e/2 + (f*x)/2)^4*sin(e /2 + (f*x)/2)^3 + 81*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^2))/(315*a^6* f*(cos(e/2 + (f*x)/2) + sin(e/2 + (f*x)/2))^9)
Time = 0.16 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-\frac {14 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}-\frac {58 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15}-\frac {18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {22}{315}}{a^{6} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \] Input:
int(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x)
Output:
(2*( - 315*tan((e + f*x)/2)**7 - 315*tan((e + f*x)/2)**6 - 945*tan((e + f* x)/2)**5 - 441*tan((e + f*x)/2)**4 - 609*tan((e + f*x)/2)**3 - 81*tan((e + f*x)/2)**2 - 99*tan((e + f*x)/2) - 11))/(315*a**6*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))