3.22 \(\int \frac{\sin ^5(x)}{(a+a \sin (x))^3} \, dx\)

Optimal. Leaf size=90 \[ \frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}+\frac{76 \sin ^2(x) \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac{13 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]

[Out]

(13*x)/(2*a^3) + (152*Cos[x])/(15*a^3) - (13*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^4)/(5*(a + a*Sin[x])^3) +
(11*Cos[x]*Sin[x]^3)/(15*a*(a + a*Sin[x])^2) + (76*Cos[x]*Sin[x]^2)/(15*(a^3 + a^3*Sin[x]))

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Rubi [A]  time = 0.208, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2765, 2977, 2734} \[ \frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}+\frac{76 \sin ^2(x) \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac{13 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^5/(a + a*Sin[x])^3,x]

[Out]

(13*x)/(2*a^3) + (152*Cos[x])/(15*a^3) - (13*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^4)/(5*(a + a*Sin[x])^3) +
(11*Cos[x]*Sin[x]^3)/(15*a*(a + a*Sin[x])^2) + (76*Cos[x]*Sin[x]^2)/(15*(a^3 + a^3*Sin[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 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^5(x)}{(a+a \sin (x))^3} \, dx &=\frac{\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}-\frac{\int \frac{\sin ^3(x) (4 a-7 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac{\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac{11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}-\frac{\int \frac{\sin ^2(x) \left (33 a^2-43 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac{\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac{11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac{76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )}-\frac{\int \sin (x) \left (152 a^3-195 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}-\frac{13 \cos (x) \sin (x)}{2 a^3}+\frac{\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac{11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac{76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.076316, size = 170, normalized size = 1.89 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (-24 \sin \left (\frac{x}{2}\right )+390 x \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+180 \cos (x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-15 \sin (2 x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-1016 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4-92 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+184 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2+12 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{60 (a \sin (x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^5/(a + a*Sin[x])^3,x]

[Out]

((Cos[x/2] + Sin[x/2])*(-24*Sin[x/2] + 12*(Cos[x/2] + Sin[x/2]) + 184*Sin[x/2]*(Cos[x/2] + Sin[x/2])^2 - 92*(C
os[x/2] + Sin[x/2])^3 - 1016*Sin[x/2]*(Cos[x/2] + Sin[x/2])^4 + 390*x*(Cos[x/2] + Sin[x/2])^5 + 180*Cos[x]*(Co
s[x/2] + Sin[x/2])^5 - 15*(Cos[x/2] + Sin[x/2])^5*Sin[2*x]))/(60*(a + a*Sin[x])^3)

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Maple [A]  time = 0.046, size = 152, normalized size = 1.7 \begin{align*}{\frac{1}{{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+6\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{1}{{a}^{3}}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+6\,{\frac{1}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+13\,{\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{4}{3\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+6\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}+12\,{\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)^5/(a+a*sin(x))^3,x)

[Out]

1/a^3/(tan(1/2*x)^2+1)^2*tan(1/2*x)^3+6/a^3/(tan(1/2*x)^2+1)^2*tan(1/2*x)^2-1/a^3/(tan(1/2*x)^2+1)^2*tan(1/2*x
)+6/a^3/(tan(1/2*x)^2+1)^2+13/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)^4-4/3/a^3/(
tan(1/2*x)+1)^3+6/a^3/(tan(1/2*x)+1)^2+12/a^3/(tan(1/2*x)+1)

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Maxima [B]  time = 2.74715, size = 340, normalized size = 3.78 \begin{align*} \frac{\frac{1325 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2673 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3805 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{4329 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{3575 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{2275 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{975 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{195 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + 304}{15 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{12 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{20 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{26 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{26 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{20 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{12 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac{5 \, a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{a^{3} \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}}\right )}} + \frac{13 \, \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)^5/(a+a*sin(x))^3,x, algorithm="maxima")

[Out]

1/15*(1325*sin(x)/(cos(x) + 1) + 2673*sin(x)^2/(cos(x) + 1)^2 + 3805*sin(x)^3/(cos(x) + 1)^3 + 4329*sin(x)^4/(
cos(x) + 1)^4 + 3575*sin(x)^5/(cos(x) + 1)^5 + 2275*sin(x)^6/(cos(x) + 1)^6 + 975*sin(x)^7/(cos(x) + 1)^7 + 19
5*sin(x)^8/(cos(x) + 1)^8 + 304)/(a^3 + 5*a^3*sin(x)/(cos(x) + 1) + 12*a^3*sin(x)^2/(cos(x) + 1)^2 + 20*a^3*si
n(x)^3/(cos(x) + 1)^3 + 26*a^3*sin(x)^4/(cos(x) + 1)^4 + 26*a^3*sin(x)^5/(cos(x) + 1)^5 + 20*a^3*sin(x)^6/(cos
(x) + 1)^6 + 12*a^3*sin(x)^7/(cos(x) + 1)^7 + 5*a^3*sin(x)^8/(cos(x) + 1)^8 + a^3*sin(x)^9/(cos(x) + 1)^9) + 1
3*arctan(sin(x)/(cos(x) + 1))/a^3

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Fricas [A]  time = 1.41183, size = 432, normalized size = 4.8 \begin{align*} \frac{15 \, \cos \left (x\right )^{5} +{\left (195 \, x + 449\right )} \cos \left (x\right )^{3} + 60 \, \cos \left (x\right )^{4} +{\left (585 \, x - 358\right )} \cos \left (x\right )^{2} - 6 \,{\left (65 \, x + 128\right )} \cos \left (x\right ) -{\left (15 \, \cos \left (x\right )^{4} -{\left (195 \, x - 404\right )} \cos \left (x\right )^{2} - 45 \, \cos \left (x\right )^{3} + 6 \,{\left (65 \, x + 127\right )} \cos \left (x\right ) + 780 \, x - 6\right )} \sin \left (x\right ) - 780 \, 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)^5/(a+a*sin(x))^3,x, algorithm="fricas")

[Out]

1/30*(15*cos(x)^5 + (195*x + 449)*cos(x)^3 + 60*cos(x)^4 + (585*x - 358)*cos(x)^2 - 6*(65*x + 128)*cos(x) - (1
5*cos(x)^4 - (195*x - 404)*cos(x)^2 - 45*cos(x)^3 + 6*(65*x + 127)*cos(x) + 780*x - 6)*sin(x) - 780*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)**5/(a+a*sin(x))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.63836, size = 119, normalized size = 1.32 \begin{align*} \frac{13 \, x}{2 \, a^{3}} + \frac{\tan \left (\frac{1}{2} \, x\right )^{3} + 6 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \tan \left (\frac{1}{2} \, x\right ) + 6}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{2} a^{3}} + \frac{2 \,{\left (90 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 405 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 665 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 445 \, \tan \left (\frac{1}{2} \, x\right ) + 107\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)^5/(a+a*sin(x))^3,x, algorithm="giac")

[Out]

13/2*x/a^3 + (tan(1/2*x)^3 + 6*tan(1/2*x)^2 - tan(1/2*x) + 6)/((tan(1/2*x)^2 + 1)^2*a^3) + 2/15*(90*tan(1/2*x)
^4 + 405*tan(1/2*x)^3 + 665*tan(1/2*x)^2 + 445*tan(1/2*x) + 107)/(a^3*(tan(1/2*x) + 1)^5)