3.21 \(\int \frac{\sin ^6(x)}{(a+a \sin (x))^3} \, dx\)

Optimal. Leaf size=101 \[ -\frac{23 x}{2 a^3}+\frac{136 \cos ^3(x)}{15 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{23 \sin ^3(x) \cos (x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac{23 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{13 \sin ^4(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]

[Out]

(-23*x)/(2*a^3) - (136*Cos[x])/(5*a^3) + (136*Cos[x]^3)/(15*a^3) + (23*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]
^5)/(5*(a + a*Sin[x])^3) + (13*Cos[x]*Sin[x]^4)/(15*a*(a + a*Sin[x])^2) + (23*Cos[x]*Sin[x]^3)/(3*(a^3 + a^3*S
in[x]))

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Rubi [A]  time = 0.226027, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2765, 2977, 2748, 2635, 8, 2633} \[ -\frac{23 x}{2 a^3}+\frac{136 \cos ^3(x)}{15 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{23 \sin ^3(x) \cos (x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac{23 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{13 \sin ^4(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^6/(a + a*Sin[x])^3,x]

[Out]

(-23*x)/(2*a^3) - (136*Cos[x])/(5*a^3) + (136*Cos[x]^3)/(15*a^3) + (23*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]
^5)/(5*(a + a*Sin[x])^3) + (13*Cos[x]*Sin[x]^4)/(15*a*(a + a*Sin[x])^2) + (23*Cos[x]*Sin[x]^3)/(3*(a^3 + a^3*S
in[x]))

Rule 2765

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^6(x)}{(a+a \sin (x))^3} \, dx &=\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}-\frac{\int \frac{\sin ^4(x) (5 a-8 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac{13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}-\frac{\int \frac{\sin ^3(x) \left (52 a^2-63 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac{13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}+\frac{23 \cos (x) \sin ^3(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac{\int \sin ^2(x) \left (345 a^3-408 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac{13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}+\frac{23 \cos (x) \sin ^3(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac{23 \int \sin ^2(x) \, dx}{a^3}+\frac{136 \int \sin ^3(x) \, dx}{5 a^3}\\ &=\frac{23 \cos (x) \sin (x)}{2 a^3}+\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac{13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}+\frac{23 \cos (x) \sin ^3(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac{23 \int 1 \, dx}{2 a^3}-\frac{136 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{5 a^3}\\ &=-\frac{23 x}{2 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{136 \cos ^3(x)}{15 a^3}+\frac{23 \cos (x) \sin (x)}{2 a^3}+\frac{\cos (x) \sin ^5(x)}{5 (a+a \sin (x))^3}+\frac{13 \cos (x) \sin ^4(x)}{15 a (a+a \sin (x))^2}+\frac{23 \cos (x) \sin ^3(x)}{3 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.102024, size = 191, normalized size = 1.89 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (24 \sin \left (\frac{x}{2}\right )-690 x \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-405 \cos (x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+5 \cos (3 x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+45 \sin (2 x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+1576 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4+112 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-224 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2-12 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{60 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^6/(a + a*Sin[x])^3,x]

[Out]

((Cos[x/2] + Sin[x/2])*(24*Sin[x/2] - 12*(Cos[x/2] + Sin[x/2]) - 224*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 + 112*(C
os[x/2] + Sin[x/2])^3 + 1576*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 - 690*x*(Cos[x/2] + Sin[x/2])^5 - 405*Cos[x]*(Co
s[x/2] + Sin[x/2])^5 + 5*Cos[3*x]*(Cos[x/2] + Sin[x/2])^5 + 45*(Cos[x/2] + Sin[x/2])^5*Sin[2*x]))/(60*(a + a*S
in[x])^3)

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Maple [A]  time = 0.052, size = 174, normalized size = 1.7 \begin{align*} -3\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{5}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-12\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{4}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-28\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+3\,{\frac{\tan \left ( x/2 \right ) }{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-{\frac{40}{3\,{a}^{3}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-23\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{3}}}-{\frac{8}{5\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+4\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{4}}}+{\frac{8}{3\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-8\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}-20\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^6/(a+a*sin(x))^3,x)

[Out]

-3/a^3/(tan(1/2*x)^2+1)^3*tan(1/2*x)^5-12/a^3/(tan(1/2*x)^2+1)^3*tan(1/2*x)^4-28/a^3/(tan(1/2*x)^2+1)^3*tan(1/
2*x)^2+3/a^3/(tan(1/2*x)^2+1)^3*tan(1/2*x)-40/3/a^3/(tan(1/2*x)^2+1)^3-23/a^3*arctan(tan(1/2*x))-8/5/a^3/(tan(
1/2*x)+1)^5+4/a^3/(tan(1/2*x)+1)^4+8/3/a^3/(tan(1/2*x)+1)^3-8/a^3/(tan(1/2*x)+1)^2-20/a^3/(tan(1/2*x)+1)

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Maxima [B]  time = 2.16743, size = 413, normalized size = 4.09 \begin{align*} -\frac{\frac{2375 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5347 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{9230 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{12622 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{13340 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{11684 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{8050 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{4370 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{1725 \, \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac{345 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + 544}{15 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{13 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{25 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{38 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{46 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{46 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{38 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{25 \, a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{13 \, a^{3} \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac{5 \, a^{3} \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac{a^{3} \sin \left (x\right )^{11}}{{\left (\cos \left (x\right ) + 1\right )}^{11}}\right )}} - \frac{23 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

-1/15*(2375*sin(x)/(cos(x) + 1) + 5347*sin(x)^2/(cos(x) + 1)^2 + 9230*sin(x)^3/(cos(x) + 1)^3 + 12622*sin(x)^4
/(cos(x) + 1)^4 + 13340*sin(x)^5/(cos(x) + 1)^5 + 11684*sin(x)^6/(cos(x) + 1)^6 + 8050*sin(x)^7/(cos(x) + 1)^7
+ 4370*sin(x)^8/(cos(x) + 1)^8 + 1725*sin(x)^9/(cos(x) + 1)^9 + 345*sin(x)^10/(cos(x) + 1)^10 + 544)/(a^3 + 5
*a^3*sin(x)/(cos(x) + 1) + 13*a^3*sin(x)^2/(cos(x) + 1)^2 + 25*a^3*sin(x)^3/(cos(x) + 1)^3 + 38*a^3*sin(x)^4/(
cos(x) + 1)^4 + 46*a^3*sin(x)^5/(cos(x) + 1)^5 + 46*a^3*sin(x)^6/(cos(x) + 1)^6 + 38*a^3*sin(x)^7/(cos(x) + 1)
^7 + 25*a^3*sin(x)^8/(cos(x) + 1)^8 + 13*a^3*sin(x)^9/(cos(x) + 1)^9 + 5*a^3*sin(x)^10/(cos(x) + 1)^10 + a^3*s
in(x)^11/(cos(x) + 1)^11) - 23*arctan(sin(x)/(cos(x) + 1))/a^3

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Fricas [A]  time = 1.55864, size = 479, normalized size = 4.74 \begin{align*} \frac{10 \, \cos \left (x\right )^{6} - 15 \, \cos \left (x\right )^{5} -{\left (345 \, x + 839\right )} \cos \left (x\right )^{3} - 140 \, \cos \left (x\right )^{4} -{\left (1035 \, x - 668\right )} \cos \left (x\right )^{2} + 6 \,{\left (115 \, x + 233\right )} \cos \left (x\right ) +{\left (10 \, \cos \left (x\right )^{5} + 25 \, \cos \left (x\right )^{4} -{\left (345 \, x - 724\right )} \cos \left (x\right )^{2} - 115 \, \cos \left (x\right )^{3} + 6 \,{\left (115 \, x + 232\right )} \cos \left (x\right ) + 1380 \, x - 6\right )} \sin \left (x\right ) + 1380 \, x + 6}{30 \,{\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} +{\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

1/30*(10*cos(x)^6 - 15*cos(x)^5 - (345*x + 839)*cos(x)^3 - 140*cos(x)^4 - (1035*x - 668)*cos(x)^2 + 6*(115*x +
233)*cos(x) + (10*cos(x)^5 + 25*cos(x)^4 - (345*x - 724)*cos(x)^2 - 115*cos(x)^3 + 6*(115*x + 232)*cos(x) + 1
380*x - 6)*sin(x) + 1380*x + 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*
cos(x) - 4*a^3)*sin(x))

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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(sin(x)**6/(a+a*sin(x))**3,x)

[Out]

Timed out

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Giac [A]  time = 2.08568, size = 134, normalized size = 1.33 \begin{align*} -\frac{23 \, x}{2 \, a^{3}} - \frac{9 \, \tan \left (\frac{1}{2} \, x\right )^{5} + 36 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 84 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 9 \, \tan \left (\frac{1}{2} \, x\right ) + 40}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3} a^{3}} - \frac{4 \,{\left (75 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 330 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 530 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 355 \, \tan \left (\frac{1}{2} \, x\right ) + 86\right )}}{15 \, a^{3}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

-23/2*x/a^3 - 1/3*(9*tan(1/2*x)^5 + 36*tan(1/2*x)^4 + 84*tan(1/2*x)^2 - 9*tan(1/2*x) + 40)/((tan(1/2*x)^2 + 1)
^3*a^3) - 4/15*(75*tan(1/2*x)^4 + 330*tan(1/2*x)^3 + 530*tan(1/2*x)^2 + 355*tan(1/2*x) + 86)/(a^3*(tan(1/2*x)
+ 1)^5)