3.441 \(\int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx\)

Optimal. Leaf size=89 \[ -\frac {47 \cos ^5(e+f x)}{315 a^2 f (a \sin (e+f x)+a)^5}-\frac {a \cos ^7(e+f x)}{18 f (a \sin (e+f x)+a)^8}+\frac {25 \cos ^5(e+f x)}{126 a f (a \sin (e+f x)+a)^6} \]

[Out]

-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

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Rubi [A]  time = 0.46, antiderivative size = 131, normalized size of antiderivative = 1.47, number of steps used = 18, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2875, 2872, 2650, 2648} \[ -\frac {47 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)}+\frac {268 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)^2}-\frac {181 \cos (e+f x)}{105 a^7 f (\sin (e+f x)+1)^3}+\frac {92 \cos (e+f x)}{63 a^7 f (\sin (e+f x)+1)^4}-\frac {4 \cos (e+f x)}{9 a^7 f (\sin (e+f x)+1)^5} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]

[Out]

(-4*Cos[e + f*x])/(9*a^7*f*(1 + Sin[e + f*x])^5) + (92*Cos[e + f*x])/(63*a^7*f*(1 + Sin[e + f*x])^4) - (181*Co
s[e + f*x])/(105*a^7*f*(1 + Sin[e + f*x])^3) + (268*Cos[e + f*x])/(315*a^7*f*(1 + Sin[e + f*x])^2) - (47*Cos[e
+ f*x])/(315*a^7*f*(1 + Sin[e + f*x]))

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]

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 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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx &=\frac {\int \sec ^8(e+f x) (a-a \sin (e+f x))^7 \tan ^2(e+f x) \, dx}{a^{14}}\\ &=\frac {\int \left (\frac {4}{a^3 (1+\sin (e+f x))^5}-\frac {12}{a^3 (1+\sin (e+f x))^4}+\frac {13}{a^3 (1+\sin (e+f x))^3}-\frac {6}{a^3 (1+\sin (e+f x))^2}+\frac {1}{a^3 (1+\sin (e+f x))}\right ) \, dx}{a^4}\\ &=\frac {\int \frac {1}{1+\sin (e+f x)} \, dx}{a^7}+\frac {4 \int \frac {1}{(1+\sin (e+f x))^5} \, dx}{a^7}-\frac {6 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{a^7}-\frac {12 \int \frac {1}{(1+\sin (e+f x))^4} \, dx}{a^7}+\frac {13 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{a^7}\\ &=-\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {12 \cos (e+f x)}{7 a^7 f (1+\sin (e+f x))^4}-\frac {13 \cos (e+f x)}{5 a^7 f (1+\sin (e+f x))^3}+\frac {2 \cos (e+f x)}{a^7 f (1+\sin (e+f x))^2}-\frac {\cos (e+f x)}{a^7 f (1+\sin (e+f x))}+\frac {16 \int \frac {1}{(1+\sin (e+f x))^4} \, dx}{9 a^7}-\frac {2 \int \frac {1}{1+\sin (e+f x)} \, dx}{a^7}-\frac {36 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{7 a^7}+\frac {26 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{5 a^7}\\ &=-\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {11 \cos (e+f x)}{7 a^7 f (1+\sin (e+f x))^3}+\frac {4 \cos (e+f x)}{15 a^7 f (1+\sin (e+f x))^2}+\frac {\cos (e+f x)}{a^7 f (1+\sin (e+f x))}+\frac {16 \int \frac {1}{(1+\sin (e+f x))^3} \, dx}{21 a^7}+\frac {26 \int \frac {1}{1+\sin (e+f x)} \, dx}{15 a^7}-\frac {72 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{35 a^7}\\ &=-\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {20 \cos (e+f x)}{21 a^7 f (1+\sin (e+f x))^2}-\frac {11 \cos (e+f x)}{15 a^7 f (1+\sin (e+f x))}+\frac {32 \int \frac {1}{(1+\sin (e+f x))^2} \, dx}{105 a^7}-\frac {24 \int \frac {1}{1+\sin (e+f x)} \, dx}{35 a^7}\\ &=-\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {268 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))^2}-\frac {\cos (e+f x)}{21 a^7 f (1+\sin (e+f x))}+\frac {32 \int \frac {1}{1+\sin (e+f x)} \, dx}{315 a^7}\\ &=-\frac {4 \cos (e+f x)}{9 a^7 f (1+\sin (e+f x))^5}+\frac {92 \cos (e+f x)}{63 a^7 f (1+\sin (e+f x))^4}-\frac {181 \cos (e+f x)}{105 a^7 f (1+\sin (e+f x))^3}+\frac {268 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))^2}-\frac {47 \cos (e+f x)}{315 a^7 f (1+\sin (e+f x))}\\ \end {align*}

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Mathematica [B]  time = 2.61, size = 293, normalized size = 3.29 \[ \frac {1890 \sin \left (e+\frac {f x}{2}\right )+1260 \sin \left (e+\frac {3 f x}{2}\right )+659400 \sin \left (2 e+\frac {3 f x}{2}\right )-303192 \sin \left (2 e+\frac {5 f x}{2}\right )-540 \sin \left (3 e+\frac {5 f x}{2}\right )-135 \sin \left (3 e+\frac {7 f x}{2}\right )-89955 \sin \left (4 e+\frac {7 f x}{2}\right )+13427 \sin \left (4 e+\frac {9 f x}{2}\right )+15 \sin \left (5 e+\frac {9 f x}{2}\right )+718830 \cos \left (e+\frac {f x}{2}\right )-467208 \cos \left (e+\frac {3 f x}{2}\right )-1260 \cos \left (2 e+\frac {3 f x}{2}\right )-540 \cos \left (2 e+\frac {5 f x}{2}\right )-179640 \cos \left (3 e+\frac {5 f x}{2}\right )+30753 \cos \left (3 e+\frac {7 f x}{2}\right )+135 \cos \left (4 e+\frac {7 f x}{2}\right )+15 \cos \left (4 e+\frac {9 f x}{2}\right )-15 \cos \left (5 e+\frac {9 f x}{2}\right )+971082 \sin \left (\frac {f x}{2}\right )+1890 \cos \left (\frac {f x}{2}\right )}{720720 a^7 f \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]

[Out]

(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*C
os[4*e + (9*f*x)/2] - 15*Cos[5*e + (9*f*x)/2] + 971082*Sin[(f*x)/2] + 1890*Sin[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)

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fricas [B]  time = 0.46, size = 243, normalized size = 2.73 \[ -\frac {47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} - 115 \, \cos \left (f x + e\right )^{3} - 265 \, \cos \left (f x + e\right )^{2} - {\left (47 \, \cos \left (f x + e\right )^{4} - 80 \, \cos \left (f x + e\right )^{3} - 195 \, \cos \left (f x + e\right )^{2} + 70 \, \cos \left (f x + e\right ) + 140\right )} \sin \left (f x + e\right ) + 70 \, \cos \left (f x + e\right ) + 140}{315 \, {\left (a^{7} f \cos \left (f x + e\right )^{5} + 5 \, a^{7} f \cos \left (f x + e\right )^{4} - 8 \, a^{7} f \cos \left (f x + e\right )^{3} - 20 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f + {\left (a^{7} f \cos \left (f x + e\right )^{4} - 4 \, a^{7} f \cos \left (f x + e\right )^{3} - 12 \, a^{7} f \cos \left (f x + e\right )^{2} + 8 \, a^{7} f \cos \left (f x + e\right ) + 16 \, a^{7} f\right )} \sin \left (f x + e\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="fricas")

[Out]

-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
*cos(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))

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giac [A]  time = 0.38, size = 106, normalized size = 1.19 \[ -\frac {4 \, {\left (210 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 126 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{315 \, a^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="giac")

[Out]

-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)

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maple [A]  time = 0.46, size = 115, normalized size = 1.29 \[ \frac {\frac {352}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {832}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {8}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {328}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}}{f \,a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x)

[Out]

8/f/a^7*(44/3/(tan(1/2*f*x+1/2*e)+1)^6-104/7/(tan(1/2*f*x+1/2*e)+1)^7+5/2/(tan(1/2*f*x+1/2*e)+1)^4-1/3/(tan(1/
2*f*x+1/2*e)+1)^3+8/(tan(1/2*f*x+1/2*e)+1)^8-41/5/(tan(1/2*f*x+1/2*e)+1)^5-16/9/(tan(1/2*f*x+1/2*e)+1)^9)

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maxima [B]  time = 0.54, size = 335, normalized size = 3.76 \[ -\frac {4 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {126 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {315 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {210 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (a^{7} + \frac {9 \, a^{7} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{7} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{7} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{7} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{7} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{7} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*sin(f*x+e)^2/(a+a*sin(f*x+e))^7,x, algorithm="maxima")

[Out]

-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/(cos(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)

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mupad [B]  time = 9.05, size = 181, normalized size = 2.03 \[ -\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+9\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-126\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+210\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{315\,a^7\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)^4*sin(e + f*x)^2)/(a + a*sin(e + f*x))^7,x)

[Out]

-(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*co
s(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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**4*sin(f*x+e)**2/(a+a*sin(f*x+e))**7,x)

[Out]

Timed out

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