3.1.21 \(\int \frac {\sin ^6(x)}{(a+a \sin (x))^3} \, dx\) [21]

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

[Out]

-23/2*x/a^3-136/5*cos(x)/a^3+136/15*cos(x)^3/a^3+23/2*cos(x)*sin(x)/a^3+1/5*cos(x)*sin(x)^5/(a+a*sin(x))^3+13/
15*cos(x)*sin(x)^4/a/(a+a*sin(x))^2+23/3*cos(x)*sin(x)^3/(a^3+a^3*sin(x))

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Rubi [A]
time = 0.16, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 = {2844, 3056, 2827, 2715, 8, 2713} \begin {gather*} -\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} \end {gather*}

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 8

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

Rule 2713

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]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2827

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 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 ^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 \text {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*}

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Mathematica [A]
time = 0.07, size = 191, 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 )-224 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+112 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+1576 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-690 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-405 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+5 \cos (3 x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+45 \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]^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 = 0.24, size = 108, normalized size = 1.07

method	result	size
default	\(\frac {-\frac {4 \left (\frac {3 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+7 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {3 \tan \left (\frac {x}{2}\right )}{4}+\frac {10}{3}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-23 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {8}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {20}{\tan \left (\frac {x}{2}\right )+1}}{a^{3}}\)	\(108\)
risch	\(-\frac {23 x}{2 a^{3}}+\frac {{\mathrm e}^{3 i x}}{24 a^{3}}-\frac {3 i {\mathrm e}^{2 i x}}{8 a^{3}}-\frac {27 \,{\mathrm e}^{i x}}{8 a^{3}}-\frac {27 \,{\mathrm e}^{-i x}}{8 a^{3}}+\frac {3 i {\mathrm e}^{-2 i x}}{8 a^{3}}+\frac {{\mathrm e}^{-3 i x}}{24 a^{3}}-\frac {2 \left (810 i {\mathrm e}^{3 i x}+225 \,{\mathrm e}^{4 i x}-1160 \,{\mathrm e}^{2 i x}-760 i {\mathrm e}^{i x}+197\right )}{15 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\)	\(117\)
norman	\(\frac {-\frac {115 x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {4409 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {460 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {15025 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2944 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {2300 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {3335 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {1564 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {920 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {6979 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {3271 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {6059 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {5509 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {9631 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {184 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {61129 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {475 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {184 x \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}-\frac {5209 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {2300 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {25829 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{15 a}-\frac {2645 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {1081 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {115 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}-\frac {23 \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{a}-\frac {115 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {2944 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {23 x \left (\tan ^{17}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {23 x}{2 a}-\frac {1564 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {460 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}-\frac {3335 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {920 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}-\frac {8533 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {544}{15 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{6} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\)	\(411\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

128/a^3*(-1/32*(3/4*tan(1/2*x)^5+3*tan(1/2*x)^4+7*tan(1/2*x)^2-3/4*tan(1/2*x)+10/3)/(tan(1/2*x)^2+1)^3-23/128*
arctan(tan(1/2*x))-1/80/(tan(1/2*x)+1)^5+1/32/(tan(1/2*x)+1)^4+1/48/(tan(1/2*x)+1)^3-1/16/(tan(1/2*x)+1)^2-5/3
2/(tan(1/2*x)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (87) = 174\).
time = 0.62, size = 306, normalized size = 3.03 \begin {gather*} -\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 {gather*}

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 = 0.35, size = 158, normalized size = 1.56 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 3288 vs. \(2 (107) = 214\).
time = 24.86, size = 3288, normalized size = 32.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-345*x*tan(x/2)**11/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**
8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*t
an(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 1725*x*tan(x/2)**10/(30*a**3*tan(x/2)**11 +
150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x
/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**
3*tan(x/2) + 30*a**3) - 4485*x*tan(x/2)**9/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**
9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*t
an(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 8625*x*tan(x/2)**8/(
30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2
)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3
*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 13110*x*tan(x/2)**7/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**1
0 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*ta
n(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3)
- 15870*x*tan(x/2)**6/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2
)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**
3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 15870*x*tan(x/2)**5/(30*a**3*tan(x/2)**1
1 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*ta
n(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*
a**3*tan(x/2) + 30*a**3) - 13110*x*tan(x/2)**4/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/
2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a*
*3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 8625*x*tan(x/2)*
*3/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan
(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*
a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 4485*x*tan(x/2)**2/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)
**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3
*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a*
*3) - 1725*x*tan(x/2)/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)
**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3
*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 345*x/(30*a**3*tan(x/2)**11 + 150*a**3*ta
n(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 138
0*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) +
30*a**3) - 690*tan(x/2)**10/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*t
an(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 7
50*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 3450*tan(x/2)**9/(30*a**3*tan(x/2)
**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3
*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 1
50*a**3*tan(x/2) + 30*a**3) - 8740*tan(x/2)**8/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/
2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a*
*3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 16100*tan(x/2)**
7/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(
x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*tan(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a
**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3) - 23368*tan(x/2)**6/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**
10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2)**6 + 1380*a**3*t
an(x/2)**5 + 1140*a**3*tan(x/2)**4 + 750*a**3*tan(x/2)**3 + 390*a**3*tan(x/2)**2 + 150*a**3*tan(x/2) + 30*a**3
) - 26680*tan(x/2)**5/(30*a**3*tan(x/2)**11 + 150*a**3*tan(x/2)**10 + 390*a**3*tan(x/2)**9 + 750*a**3*tan(x/2)
**8 + 1140*a**3*tan(x/2)**7 + 1380*a**3*tan(x/2...

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Giac [A]
time = 0.47, size = 99, normalized size = 0.98 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 7.02, size = 110, normalized size = 1.09 \begin {gather*} -\frac {23\,x}{2\,a^3}-\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}+115\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9+\frac {874\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{3}+\frac {1610\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{3}+\frac {11684\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{15}+\frac {2668\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {12622\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{15}+\frac {1846\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {5347\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{15}+\frac {475\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {544}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,{\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)^6/(a + a*sin(x))^3,x)

[Out]

- (23*x)/(2*a^3) - ((475*tan(x/2))/3 + (5347*tan(x/2)^2)/15 + (1846*tan(x/2)^3)/3 + (12622*tan(x/2)^4)/15 + (2
668*tan(x/2)^5)/3 + (11684*tan(x/2)^6)/15 + (1610*tan(x/2)^7)/3 + (874*tan(x/2)^8)/3 + 115*tan(x/2)^9 + 23*tan
(x/2)^10 + 544/15)/(a^3*(tan(x/2)^2 + 1)^3*(tan(x/2) + 1)^5)

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