3.1.22 \(\int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx\) [22]

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

[Out]

13/2*x/a^3+152/15*cos(x)/a^3-13/2*cos(x)*sin(x)/a^3+1/5*cos(x)*sin(x)^4/(a+a*sin(x))^3+11/15*cos(x)*sin(x)^3/a
/(a+a*sin(x))^2+76/15*cos(x)*sin(x)^2/(a^3+a^3*sin(x))

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Rubi [A]
time = 0.15, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 = {2844, 3056, 2813} \begin {gather*} \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} \end {gather*}

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 2813

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]

Rule 2844

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 3056

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])

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*}

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Mathematica [A]
time = 0.06, size = 170, normalized size = 1.89 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-24 \sin \left (\frac {x}{2}\right )+12 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+184 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-92 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-1016 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+390 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+180 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-15 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \sin (2 x)\right )}{60 (a+a \sin (x))^3} \end {gather*}

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 = 0.22, size = 100, normalized size = 1.11

method	result	size
risch	\(\frac {13 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i x}}{8 a^{3}}+\frac {3 \,{\mathrm e}^{i x}}{2 a^{3}}+\frac {3 \,{\mathrm e}^{-i x}}{2 a^{3}}-\frac {i {\mathrm e}^{-2 i x}}{8 a^{3}}+\frac {70 i {\mathrm e}^{3 i x}+20 \,{\mathrm e}^{4 i x}-\frac {194 i {\mathrm e}^{i x}}{3}-\frac {298 \,{\mathrm e}^{2 i x}}{3}+\frac {254}{15}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\)	\(97\)
default	\(\frac {\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {\tan \left (\frac {x}{2}\right )}{4}+\frac {3}{2}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+13 \arctan \left (\tan \left (\frac {x}{2}\right )\right )+\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {6}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {12}{\tan \left (\frac {x}{2}\right )+1}}{a^{3}}\)	\(100\)
norman	\(\frac {\frac {4717 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {455 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3491 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {2015 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1755 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {2015 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1313 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {845 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {239 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {1556 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {884 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {1512 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}+\frac {3793 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {195 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3913 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {265 \tan \left (\frac {x}{2}\right )}{3 a}+\frac {13 x \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {4888 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {845 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {572 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {65 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}+\frac {13 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}+\frac {65 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {1313 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {13 x}{2 a}+\frac {455 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {65 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1755 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {195 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1300 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {304}{15 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{5} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)

[Out]

64/a^3*(1/16*(1/4*tan(1/2*x)^3+3/2*tan(1/2*x)^2-1/4*tan(1/2*x)+3/2)/(tan(1/2*x)^2+1)^2+13/64*arctan(tan(1/2*x)
)+1/40/(tan(1/2*x)+1)^5-1/16/(tan(1/2*x)+1)^4-1/48/(tan(1/2*x)+1)^3+3/32/(tan(1/2*x)+1)^2+3/16/(tan(1/2*x)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (78) = 156\).
time = 0.57, size = 252, normalized size = 2.80 \begin {gather*} \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 {gather*}

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 = 0.34, size = 145, normalized size = 1.61 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (95) = 190\).
time = 15.53, size = 2259, normalized size = 25.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**5/(a+a*sin(x))**3,x)

[Out]

195*x*tan(x/2)**9/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 +
780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2)
+ 30*a**3) + 975*x*tan(x/2)**8/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*t
an(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a
**3*tan(x/2) + 30*a**3) + 2340*x*tan(x/2)**7/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**
7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x
/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 3900*x*tan(x/2)**6/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a
**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 +
360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 5070*x*tan(x/2)**5/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x
/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*
tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 5070*x*tan(x/2)**4/(30*a**3*tan(x/2)**9 +
150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)*
*4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 3900*x*tan(x/2)**3/(30*a**3*
tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*
a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 2340*x*tan(x/2
)**2/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(
x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) +
975*x*tan(x/2)/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780
*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 3
0*a**3) + 195*x/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 78
0*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) +
30*a**3) + 390*tan(x/2)**8/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x
/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*
tan(x/2) + 30*a**3) + 1950*tan(x/2)**7/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 60
0*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2
+ 150*a**3*tan(x/2) + 30*a**3) + 4550*tan(x/2)**6/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(
x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3
*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 7150*tan(x/2)**5/(30*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 3
60*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**
3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 8658*tan(x/2)**4/(30*a**3*tan(x/2)**9 + 150*a**3*tan
(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)**4 + 600*a**
3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 7610*tan(x/2)**3/(30*a**3*tan(x/2)**9 +
150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a**3*tan(x/2)*
*4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 5346*tan(x/2)**2/(30*a**3*ta
n(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5 + 780*a*
*3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 2650*tan(x/2)/(3
0*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5
+ 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) + 608/(30
*a**3*tan(x/2)**9 + 150*a**3*tan(x/2)**8 + 360*a**3*tan(x/2)**7 + 600*a**3*tan(x/2)**6 + 780*a**3*tan(x/2)**5
+ 780*a**3*tan(x/2)**4 + 600*a**3*tan(x/2)**3 + 360*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3)

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Giac [A]
time = 0.56, size = 88, normalized size = 0.98 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 6.67, size = 93, normalized size = 1.03 \begin {gather*} \frac {13\,x}{2\,a^3}+\frac {13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+65\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+\frac {455\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{3}+\frac {715\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {1443\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{5}+\frac {761\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {891\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {265\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {304}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(a + a*sin(x))^3,x)

[Out]

(13*x)/(2*a^3) + ((265*tan(x/2))/3 + (891*tan(x/2)^2)/5 + (761*tan(x/2)^3)/3 + (1443*tan(x/2)^4)/5 + (715*tan(
x/2)^5)/3 + (455*tan(x/2)^6)/3 + 65*tan(x/2)^7 + 13*tan(x/2)^8 + 304/15)/(a^3*(tan(x/2)^2 + 1)^2*(tan(x/2) + 1
)^5)

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