\(\int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx\) [442]

Optimal result

Integrand size = 29, antiderivative size = 157 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=\frac {4 \cos (e+f x)}{11 a^8 f (1+\sin (e+f x))^6}-\frac {52 \cos (e+f x)}{33 a^8 f (1+\sin (e+f x))^5}+\frac {617 \cos (e+f x)}{231 a^8 f (1+\sin (e+f x))^4}-\frac {846 \cos (e+f x)}{385 a^8 f (1+\sin (e+f x))^3}+\frac {1003 \cos (e+f x)}{1155 a^8 f (1+\sin (e+f x))^2}-\frac {152 \cos (e+f x)}{1155 a^8 f (1+\sin (e+f x))} \]

[Out]

4/11*cos(f*x+e)/a^8/f/(1+sin(f*x+e))^6-52/33*cos(f*x+e)/a^8/f/(1+sin(f*x+e))^5+617/231*cos(f*x+e)/a^8/f/(1+sin
(f*x+e))^4-846/385*cos(f*x+e)/a^8/f/(1+sin(f*x+e))^3+1003/1155*cos(f*x+e)/a^8/f/(1+sin(f*x+e))^2-152/1155*cos(
f*x+e)/a^8/f/(1+sin(f*x+e))

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2954, 2951, 2729, 2727} \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=-\frac {152 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)}+\frac {1003 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)^2}-\frac {846 \cos (e+f x)}{385 a^8 f (\sin (e+f x)+1)^3}+\frac {617 \cos (e+f x)}{231 a^8 f (\sin (e+f x)+1)^4}-\frac {52 \cos (e+f x)}{33 a^8 f (\sin (e+f x)+1)^5}+\frac {4 \cos (e+f x)}{11 a^8 f (\sin (e+f x)+1)^6} \]

[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 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 2951

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 2954

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 3.80 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=-\frac {-486024 \cos \left (e+\frac {f x}{2}\right )+351450 \cos \left (e+\frac {3 f x}{2}\right )+180015 \cos \left (3 e+\frac {5 f x}{2}\right )-63580 \cos \left (3 e+\frac {7 f x}{2}\right )-15004 \cos \left (5 e+\frac {9 f x}{2}\right )+1975 \cos \left (5 e+\frac {11 f x}{2}\right )-425964 \sin \left (\frac {f x}{2}\right )-299970 \sin \left (2 e+\frac {3 f x}{2}\right )+145695 \sin \left (2 e+\frac {5 f x}{2}\right )+44990 \sin \left (4 e+\frac {7 f x}{2}\right )-6710 \sin \left (4 e+\frac {9 f x}{2}\right )+\sin \left (6 e+\frac {11 f x}{2}\right )}{240240 a^8 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}} \]

[In]

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

[Out]

-1/240240*(-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])/(a^8*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^11)

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {-\frac {4}{1155}-\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{105}-\frac {4 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}-\frac {44 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {36 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-4 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {20 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}}{f \,a^{8} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}\)	\(113\)
derivativedivides	\(\frac {\frac {176}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{10}}+\frac {896}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {384}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {2064}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{8}}\)	\(130\)
default	\(\frac {\frac {176}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{10}}+\frac {896}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {384}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {2064}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{8}}\)	\(130\)
risch	\(-\frac {2 \left (3465 i {\mathrm e}^{9 i \left (f x +e \right )}+1155 \,{\mathrm e}^{10 i \left (f x +e \right )}-23100 i {\mathrm e}^{7 i \left (f x +e \right )}-13860 \,{\mathrm e}^{8 i \left (f x +e \right )}+32802 i {\mathrm e}^{5 i \left (f x +e \right )}+37422 \,{\mathrm e}^{6 i \left (f x +e \right )}-11220 i {\mathrm e}^{3 i \left (f x +e \right )}-27060 \,{\mathrm e}^{4 i \left (f x +e \right )}+517 i {\mathrm e}^{i \left (f x +e \right )}+4895 \,{\mathrm e}^{2 i \left (f x +e \right )}-152\right )}{1155 f \,a^{8} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{11}}\)	\(140\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (145) = 290\).

Time = 0.25 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=\frac {152 \, \cos \left (f x + e\right )^{6} - 243 \, \cos \left (f x + e\right )^{5} - 745 \, \cos \left (f x + e\right )^{4} + 455 \, \cos \left (f x + e\right )^{3} + 1015 \, \cos \left (f x + e\right )^{2} + {\left (152 \, \cos \left (f x + e\right )^{5} + 395 \, \cos \left (f x + e\right )^{4} - 350 \, \cos \left (f x + e\right )^{3} - 805 \, \cos \left (f x + e\right )^{2} + 210 \, \cos \left (f x + e\right ) + 420\right )} \sin \left (f x + e\right ) - 210 \, \cos \left (f x + e\right ) - 420}{1155 \, {\left (a^{8} f \cos \left (f x + e\right )^{6} - 5 \, a^{8} f \cos \left (f x + e\right )^{5} - 18 \, a^{8} f \cos \left (f x + e\right )^{4} + 20 \, a^{8} f \cos \left (f x + e\right )^{3} + 48 \, a^{8} f \cos \left (f x + e\right )^{2} - 16 \, a^{8} f \cos \left (f x + e\right ) - 32 \, a^{8} f - {\left (a^{8} f \cos \left (f x + e\right )^{5} + 6 \, a^{8} f \cos \left (f x + e\right )^{4} - 12 \, a^{8} f \cos \left (f x + e\right )^{3} - 32 \, a^{8} f \cos \left (f x + e\right )^{2} + 16 \, a^{8} f \cos \left (f x + e\right ) + 32 \, a^{8} f\right )} \sin \left (f x + e\right )\right )}} \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (145) = 290\).

Time = 0.23 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.55 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=-\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} \]

[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)

Giac [A] (verification not implemented)

none

Time = 1.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=-\frac {4 \, {\left (1155 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 2079 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 2541 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 825 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 165 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 55 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{1155 \, a^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{11}} \]

[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)

Mupad [B] (verification not implemented)

Time = 10.47 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx=-\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+11\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+55\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+165\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-825\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2541\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-2079\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+1155\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{1155\,a^8\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^{11}} \]

[In]

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

[Out]

-(4*cos(e/2 + (f*x)/2)^4*(cos(e/2 + (f*x)/2)^7 + 1155*sin(e/2 + (f*x)/2)^7 - 2079*cos(e/2 + (f*x)/2)*sin(e/2 +
(f*x)/2)^6 + 11*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2) + 2541*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^5 - 82
5*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^4 + 165*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^3 + 55*cos(e/2 + (f*
x)/2)^5*sin(e/2 + (f*x)/2)^2))/(1155*a^8*f*(cos(e/2 + (f*x)/2) + sin(e/2 + (f*x)/2))^11)