\(\int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx\) [321]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [C] (verified)
   Fricas [A] (verification not implemented)
   Sympy [B] (verification not implemented)
   Maxima [B] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 27, antiderivative size = 144 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}+\frac {4 \cos (e+f x)}{315 f \left (a^6+a^6 \sin (e+f x)\right )} \]

[Out]

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*si
n(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))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2936, 2829, 2729, 2727} \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4 \cos (e+f x)}{315 f \left (a^6 \sin (e+f x)+a^6\right )}+\frac {4 \cos (e+f x)}{315 f \left (a^3 \sin (e+f x)+a^3\right )^2}+\frac {2 \cos (e+f x)}{105 f \left (a^2 \sin (e+f x)+a^2\right )^3}-\frac {19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5} \]

[In]

Int[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]

[Out]

(2*Cos[e + f*x])/(9*a*f*(a + a*Sin[e + f*x])^5) - (19*Cos[e + f*x])/(63*a^2*f*(a + a*Sin[e + f*x])^4) + (2*Cos
[e + f*x])/(105*f*(a^2 + a^2*Sin[e + f*x])^3) + (4*Cos[e + f*x])/(315*f*(a^3 + a^3*Sin[e + f*x])^2) + (4*Cos[e
 + f*x])/(315*f*(a^6 + a^6*Sin[e + f*x]))

Rule 2727

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]

Rule 2729

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] + Dist[(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]

Rule 2829

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] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(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)]

Rule 2936

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] + Dist[
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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {\int \frac {-10 a+9 a \sin (e+f x)}{(a+a \sin (e+f x))^4} \, dx}{9 a^3} \\ & = \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 \int \frac {1}{(a+a \sin (e+f x))^3} \, dx}{21 a^3} \\ & = \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 \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{105 a^4} \\ & = \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 \int \frac {1}{a+a \sin (e+f x)} \, dx}{315 a^5} \\ & = \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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {378 \cos \left (e+\frac {f x}{2}\right )+210 \cos \left (e+\frac {3 f x}{2}\right )-108 \cos \left (3 e+\frac {5 f x}{2}\right )+225 \cos \left (3 e+\frac {7 f x}{2}\right )+3 \cos \left (5 e+\frac {9 f x}{2}\right )+3150 \sin \left (\frac {f x}{2}\right )+2562 \sin \left (2 e+\frac {3 f x}{2}\right )-900 \sin \left (2 e+\frac {5 f x}{2}\right )-27 \sin \left (4 e+\frac {7 f x}{2}\right )+25 \sin \left (4 e+\frac {9 f x}{2}\right )}{13860 a^6 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]

[In]

Integrate[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]

[Out]

-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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (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 \(\frac {-\frac {22}{315}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {18 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}-6 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {58 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {14 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}}{f \,a^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}\) \(113\)
derivativedivides \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{6}}\) \(130\)
default \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{6}}\) \(130\)
norman \(\frac {-\frac {22}{315 a f}-\frac {2 \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6 \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {18 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {242 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{315 f a}-\frac {174 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {274 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 f a}-\frac {646 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {862 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {1202 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 f a}-\frac {1762 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {8342 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {9442 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {17494 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {23858 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {28786 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}\) \(325\)

[In]

int(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

-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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4 \, \cos \left (f x + e\right )^{5} - 16 \, \cos \left (f x + e\right )^{4} - 50 \, \cos \left (f x + e\right )^{3} - 65 \, \cos \left (f x + e\right )^{2} - {\left (4 \, \cos \left (f x + e\right )^{4} + 20 \, \cos \left (f x + e\right )^{3} - 30 \, \cos \left (f x + e\right )^{2} + 35 \, \cos \left (f x + e\right ) + 70\right )} \sin \left (f x + e\right ) + 35 \, \cos \left (f x + e\right ) + 70}{315 \, {\left (a^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{6} f \cos \left (f x + e\right )^{4} - 8 \, a^{6} f \cos \left (f x + e\right )^{3} - 20 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f + {\left (a^{6} f \cos \left (f x + e\right )^{4} - 4 \, a^{6} f \cos \left (f x + e\right )^{3} - 12 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

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*c
os(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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (131) = 262\).

Time = 43.59 (sec) , antiderivative size = 1501, normalized size of antiderivative = 10.42 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\text {Too large to display} \]

[In]

integrate(cos(f*x+e)**2*sin(f*x+e)/(a+a*sin(f*x+e))**6,x)

[Out]

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 + 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) - 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*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) - 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 + 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) - 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/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) - 162*tan(e/2
 + f*x/2)**2/(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) - 198*tan(e/2 + f*x/2)/(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) - 22/(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), Ne(f, 0)), (x*sin(e)*cos(e)**2/(a*sin(e) + a)**6, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (134) = 268\).

Time = 0.23 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (\frac {99 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {81 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {609 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {945 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {315 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 11\right )}}{315 \, {\left (a^{6} + \frac {9 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{6} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{6} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \]

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-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)/(co
s(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 + 12
6*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)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 945 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 609 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 81 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 99 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11\right )}}{315 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \]

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-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*(t
an(1/2*f*x + 1/2*e) + 1)^9)

Mupad [B] (verification not implemented)

Time = 9.48 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+99\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+81\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+609\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+315\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{315\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \]

[In]

int((cos(e + f*x)^2*sin(e + f*x))/(a + a*sin(e + f*x))^6,x)

[Out]

-(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 + 44
1*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)