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]

-(a*Cos[e + f*x]^7)/(18*f*(a + a*Sin[e + f*x])^8) + (25*Cos[e + f*x]^5)/(126*a*f*(a + a*Sin[e + f*x])^6) - (47
*Cos[e + f*x]^5)/(315*a^2*f*(a + a*Sin[e + f*x])^5)

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Rubi [A]  time = 0.458959, 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 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 ^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*}

Mathematica [B]  time = 2.81116, 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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Maple [A]  time = 0.132, size = 115, normalized size = 1.3 \begin{align*} 8\,{\frac{1}{f{a}^{7}} \left ( 5/2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-4}-{\frac{41}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{5}}}+{\frac{44}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{6}}}+8\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-8}-1/3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-3}-{\frac{104}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{7}}}-{\frac{16}{9\, \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)^2/(a+a*sin(f*x+e))^7,x)

[Out]

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

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Maxima [B]  time = 1.07027, size = 452, normalized size = 5.08 \begin{align*} -\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} \end{align*}

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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Fricas [B]  time = 1.06934, size = 636, normalized size = 7.15 \begin{align*} -\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 )}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.40673, size = 143, normalized size = 1.61 \begin{align*} -\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}} \end{align*}

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)