3.442 \(\int \frac{\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx\)

Optimal. Leaf size=157 \[ -\frac{152 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)}+\frac{1003 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)^2}-\frac{846 \cos (e+f x)}{385 a^8 f (\sin (e+f x)+1)^3}+\frac{617 \cos (e+f x)}{231 a^8 f (\sin (e+f x)+1)^4}-\frac{52 \cos (e+f x)}{33 a^8 f (\sin (e+f x)+1)^5}+\frac{4 \cos (e+f x)}{11 a^8 f (\sin (e+f x)+1)^6} \]

[Out]

(4*Cos[e + f*x])/(11*a^8*f*(1 + Sin[e + f*x])^6) - (52*Cos[e + f*x])/(33*a^8*f*(1 + Sin[e + f*x])^5) + (617*Co
s[e + f*x])/(231*a^8*f*(1 + Sin[e + f*x])^4) - (846*Cos[e + f*x])/(385*a^8*f*(1 + Sin[e + f*x])^3) + (1003*Cos
[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x])^2) - (152*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x]))

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Rubi [A]  time = 0.56958, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2875, 2872, 2650, 2648} \[ -\frac{152 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)}+\frac{1003 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)^2}-\frac{846 \cos (e+f x)}{385 a^8 f (\sin (e+f x)+1)^3}+\frac{617 \cos (e+f x)}{231 a^8 f (\sin (e+f x)+1)^4}-\frac{52 \cos (e+f x)}{33 a^8 f (\sin (e+f x)+1)^5}+\frac{4 \cos (e+f x)}{11 a^8 f (\sin (e+f x)+1)^6} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]^4*Sin[e + f*x]^3)/(a + a*Sin[e + f*x])^8,x]

[Out]

(4*Cos[e + f*x])/(11*a^8*f*(1 + Sin[e + f*x])^6) - (52*Cos[e + f*x])/(33*a^8*f*(1 + Sin[e + f*x])^5) + (617*Co
s[e + f*x])/(231*a^8*f*(1 + Sin[e + f*x])^4) - (846*Cos[e + f*x])/(385*a^8*f*(1 + Sin[e + f*x])^3) + (1003*Cos
[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x])^2) - (152*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x]))

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(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]

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(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] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 2650

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 2648

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]

Rubi steps

\begin{align*} \int \frac{\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx &=\frac{\int \sec ^9(e+f x) (a-a \sin (e+f x))^8 \tan ^3(e+f x) \, dx}{a^{16}}\\ &=\frac{\int \left (-\frac{4}{a^4 (1+\sin (e+f x))^6}+\frac{16}{a^4 (1+\sin (e+f x))^5}-\frac{25}{a^4 (1+\sin (e+f x))^4}+\frac{19}{a^4 (1+\sin (e+f x))^3}-\frac{7}{a^4 (1+\sin (e+f x))^2}+\frac{1}{a^4 (1+\sin (e+f x))}\right ) \, dx}{a^4}\\ &=\frac{\int \frac{1}{1+\sin (e+f x)} \, dx}{a^8}-\frac{4 \int \frac{1}{(1+\sin (e+f x))^6} \, dx}{a^8}-\frac{7 \int \frac{1}{(1+\sin (e+f x))^2} \, dx}{a^8}+\frac{16 \int \frac{1}{(1+\sin (e+f x))^5} \, dx}{a^8}+\frac{19 \int \frac{1}{(1+\sin (e+f x))^3} \, dx}{a^8}-\frac{25 \int \frac{1}{(1+\sin (e+f x))^4} \, dx}{a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{16 \cos (e+f x)}{9 a^8 f (1+\sin (e+f x))^5}+\frac{25 \cos (e+f x)}{7 a^8 f (1+\sin (e+f x))^4}-\frac{19 \cos (e+f x)}{5 a^8 f (1+\sin (e+f x))^3}+\frac{7 \cos (e+f x)}{3 a^8 f (1+\sin (e+f x))^2}-\frac{\cos (e+f x)}{a^8 f (1+\sin (e+f x))}-\frac{20 \int \frac{1}{(1+\sin (e+f x))^5} \, dx}{11 a^8}-\frac{7 \int \frac{1}{1+\sin (e+f x)} \, dx}{3 a^8}+\frac{64 \int \frac{1}{(1+\sin (e+f x))^4} \, dx}{9 a^8}+\frac{38 \int \frac{1}{(1+\sin (e+f x))^2} \, dx}{5 a^8}-\frac{75 \int \frac{1}{(1+\sin (e+f x))^3} \, dx}{7 a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac{23 \cos (e+f x)}{9 a^8 f (1+\sin (e+f x))^4}-\frac{58 \cos (e+f x)}{35 a^8 f (1+\sin (e+f x))^3}-\frac{\cos (e+f x)}{5 a^8 f (1+\sin (e+f x))^2}+\frac{4 \cos (e+f x)}{3 a^8 f (1+\sin (e+f x))}-\frac{80 \int \frac{1}{(1+\sin (e+f x))^4} \, dx}{99 a^8}+\frac{38 \int \frac{1}{1+\sin (e+f x)} \, dx}{15 a^8}+\frac{64 \int \frac{1}{(1+\sin (e+f x))^3} \, dx}{21 a^8}-\frac{30 \int \frac{1}{(1+\sin (e+f x))^2} \, dx}{7 a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac{617 \cos (e+f x)}{231 a^8 f (1+\sin (e+f x))^4}-\frac{34 \cos (e+f x)}{15 a^8 f (1+\sin (e+f x))^3}+\frac{43 \cos (e+f x)}{35 a^8 f (1+\sin (e+f x))^2}-\frac{6 \cos (e+f x)}{5 a^8 f (1+\sin (e+f x))}-\frac{80 \int \frac{1}{(1+\sin (e+f x))^3} \, dx}{231 a^8}+\frac{128 \int \frac{1}{(1+\sin (e+f x))^2} \, dx}{105 a^8}-\frac{10 \int \frac{1}{1+\sin (e+f x)} \, dx}{7 a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac{617 \cos (e+f x)}{231 a^8 f (1+\sin (e+f x))^4}-\frac{846 \cos (e+f x)}{385 a^8 f (1+\sin (e+f x))^3}+\frac{37 \cos (e+f x)}{45 a^8 f (1+\sin (e+f x))^2}+\frac{8 \cos (e+f x)}{35 a^8 f (1+\sin (e+f x))}-\frac{32 \int \frac{1}{(1+\sin (e+f x))^2} \, dx}{231 a^8}+\frac{128 \int \frac{1}{1+\sin (e+f x)} \, dx}{315 a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac{617 \cos (e+f x)}{231 a^8 f (1+\sin (e+f x))^4}-\frac{846 \cos (e+f x)}{385 a^8 f (1+\sin (e+f x))^3}+\frac{1003 \cos (e+f x)}{1155 a^8 f (1+\sin (e+f x))^2}-\frac{8 \cos (e+f x)}{45 a^8 f (1+\sin (e+f x))}-\frac{32 \int \frac{1}{1+\sin (e+f x)} \, dx}{693 a^8}\\ &=\frac{4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac{52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac{617 \cos (e+f x)}{231 a^8 f (1+\sin (e+f x))^4}-\frac{846 \cos (e+f x)}{385 a^8 f (1+\sin (e+f x))^3}+\frac{1003 \cos (e+f x)}{1155 a^8 f (1+\sin (e+f x))^2}-\frac{152 \cos (e+f x)}{1155 a^8 f (1+\sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 3.07504, size = 195, normalized size = 1.24 \[ -\frac{-299970 \sin \left (2 e+\frac{3 f x}{2}\right )+145695 \sin \left (2 e+\frac{5 f x}{2}\right )+44990 \sin \left (4 e+\frac{7 f x}{2}\right )-6710 \sin \left (4 e+\frac{9 f x}{2}\right )+\sin \left (6 e+\frac{11 f x}{2}\right )-486024 \cos \left (e+\frac{f x}{2}\right )+351450 \cos \left (e+\frac{3 f x}{2}\right )+180015 \cos \left (3 e+\frac{5 f x}{2}\right )-63580 \cos \left (3 e+\frac{7 f x}{2}\right )-15004 \cos \left (5 e+\frac{9 f x}{2}\right )+1975 \cos \left (5 e+\frac{11 f x}{2}\right )-425964 \sin \left (\frac{f x}{2}\right )}{240240 a^8 f \left (\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]^4*Sin[e + f*x]^3)/(a + a*Sin[e + f*x])^8,x]

[Out]

-(-486024*Cos[e + (f*x)/2] + 351450*Cos[e + (3*f*x)/2] + 180015*Cos[3*e + (5*f*x)/2] - 63580*Cos[3*e + (7*f*x)
/2] - 15004*Cos[5*e + (9*f*x)/2] + 1975*Cos[5*e + (11*f*x)/2] - 425964*Sin[(f*x)/2] - 299970*Sin[2*e + (3*f*x)
/2] + 145695*Sin[2*e + (5*f*x)/2] + 44990*Sin[4*e + (7*f*x)/2] - 6710*Sin[4*e + (9*f*x)/2] + Sin[6*e + (11*f*x
)/2])/(240240*a^8*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^11)

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Maple [A]  time = 0.156, size = 130, normalized size = 0.8 \begin{align*} 16\,{\frac{1}{f{a}^{8}} \left ({\frac{11}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{5}}}-1/4\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-4}-24\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-8}-8\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-10}+{\frac{129}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{7}}}+{\frac{16}{11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{11}}}-17/2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-6}+{\frac{56}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^4*sin(f*x+e)^3/(a+a*sin(f*x+e))^8,x)

[Out]

16/f/a^8*(11/5/(tan(1/2*f*x+1/2*e)+1)^5-1/4/(tan(1/2*f*x+1/2*e)+1)^4-24/(tan(1/2*f*x+1/2*e)+1)^8-8/(tan(1/2*f*
x+1/2*e)+1)^10+129/7/(tan(1/2*f*x+1/2*e)+1)^7+16/11/(tan(1/2*f*x+1/2*e)+1)^11-17/2/(tan(1/2*f*x+1/2*e)+1)^6+56
/3/(tan(1/2*f*x+1/2*e)+1)^9)

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Maxima [B]  time = 1.24958, size = 541, normalized size = 3.45 \begin{align*} -\frac{4 \,{\left (\frac{11 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{55 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{165 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{825 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{2541 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac{2079 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{1155 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 1\right )}}{1155 \,{\left (a^{8} + \frac{11 \, a^{8} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{55 \, a^{8} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{165 \, a^{8} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{330 \, a^{8} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{462 \, a^{8} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{462 \, a^{8} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{330 \, a^{8} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac{165 \, a^{8} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{55 \, a^{8} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac{11 \, a^{8} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac{a^{8} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^3/(a+a*sin(f*x+e))^8,x, algorithm="maxima")

[Out]

-4/1155*(11*sin(f*x + e)/(cos(f*x + e) + 1) + 55*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 165*sin(f*x + e)^3/(cos
(f*x + e) + 1)^3 - 825*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2541*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 2079*s
in(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1155*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1)/((a^8 + 11*a^8*sin(f*x + e)
/(cos(f*x + e) + 1) + 55*a^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 165*a^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3
+ 330*a^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 462*a^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*a^8*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 + 330*a^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*a^8*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 + 55*a^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*a^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + a^8*s
in(f*x + e)^11/(cos(f*x + e) + 1)^11)*f)

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Fricas [B]  time = 1.05793, size = 775, normalized size = 4.94 \begin{align*} \frac{152 \, \cos \left (f x + e\right )^{6} - 243 \, \cos \left (f x + e\right )^{5} - 745 \, \cos \left (f x + e\right )^{4} + 455 \, \cos \left (f x + e\right )^{3} + 1015 \, \cos \left (f x + e\right )^{2} +{\left (152 \, \cos \left (f x + e\right )^{5} + 395 \, \cos \left (f x + e\right )^{4} - 350 \, \cos \left (f x + e\right )^{3} - 805 \, \cos \left (f x + e\right )^{2} + 210 \, \cos \left (f x + e\right ) + 420\right )} \sin \left (f x + e\right ) - 210 \, \cos \left (f x + e\right ) - 420}{1155 \,{\left (a^{8} f \cos \left (f x + e\right )^{6} - 5 \, a^{8} f \cos \left (f x + e\right )^{5} - 18 \, a^{8} f \cos \left (f x + e\right )^{4} + 20 \, a^{8} f \cos \left (f x + e\right )^{3} + 48 \, a^{8} f \cos \left (f x + e\right )^{2} - 16 \, a^{8} f \cos \left (f x + e\right ) - 32 \, a^{8} f -{\left (a^{8} f \cos \left (f x + e\right )^{5} + 6 \, a^{8} f \cos \left (f x + e\right )^{4} - 12 \, a^{8} f \cos \left (f x + e\right )^{3} - 32 \, a^{8} f \cos \left (f x + e\right )^{2} + 16 \, a^{8} f \cos \left (f x + e\right ) + 32 \, a^{8} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^3/(a+a*sin(f*x+e))^8,x, algorithm="fricas")

[Out]

1/1155*(152*cos(f*x + e)^6 - 243*cos(f*x + e)^5 - 745*cos(f*x + e)^4 + 455*cos(f*x + e)^3 + 1015*cos(f*x + e)^
2 + (152*cos(f*x + e)^5 + 395*cos(f*x + e)^4 - 350*cos(f*x + e)^3 - 805*cos(f*x + e)^2 + 210*cos(f*x + e) + 42
0)*sin(f*x + e) - 210*cos(f*x + e) - 420)/(a^8*f*cos(f*x + e)^6 - 5*a^8*f*cos(f*x + e)^5 - 18*a^8*f*cos(f*x +
e)^4 + 20*a^8*f*cos(f*x + e)^3 + 48*a^8*f*cos(f*x + e)^2 - 16*a^8*f*cos(f*x + e) - 32*a^8*f - (a^8*f*cos(f*x +
e)^5 + 6*a^8*f*cos(f*x + e)^4 - 12*a^8*f*cos(f*x + e)^3 - 32*a^8*f*cos(f*x + e)^2 + 16*a^8*f*cos(f*x + e) + 3
2*a^8*f)*sin(f*x + e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**4*sin(f*x+e)**3/(a+a*sin(f*x+e))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.44066, size = 162, normalized size = 1.03 \begin{align*} -\frac{4 \,{\left (1155 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 2079 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 2541 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 825 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 165 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 55 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 11 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}}{1155 \, a^{8} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^3/(a+a*sin(f*x+e))^8,x, algorithm="giac")

[Out]

-4/1155*(1155*tan(1/2*f*x + 1/2*e)^7 - 2079*tan(1/2*f*x + 1/2*e)^6 + 2541*tan(1/2*f*x + 1/2*e)^5 - 825*tan(1/2
*f*x + 1/2*e)^4 + 165*tan(1/2*f*x + 1/2*e)^3 + 55*tan(1/2*f*x + 1/2*e)^2 + 11*tan(1/2*f*x + 1/2*e) + 1)/(a^8*f
*(tan(1/2*f*x + 1/2*e) + 1)^11)