3.5.24 \(\int \frac {\sin (5 x)}{(5 \cos ^2(x)+9 \sin ^2(x))^{5/2}} \, dx\) [424]

Optimal. Leaf size=48 \[ -\frac {1}{2} \sin ^{-1}\left (\frac {2 \cos (x)}{3}\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}} \]

[Out]

-1/2*arcsin(2/3*cos(x))-55/27*cos(x)/(9-4*cos(x)^2)^(3/2)+295/243*cos(x)/(9-4*cos(x)^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1171, 393, 222} \begin {gather*} -\frac {1}{2} \text {ArcSin}\left (\frac {2 \cos (x)}{3}\right )+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]

[Out]

-1/2*ArcSin[(2*Cos[x])/3] - (55*Cos[x])/(27*(9 - 4*Cos[x]^2)^(3/2)) + (295*Cos[x])/(243*Sqrt[9 - 4*Cos[x]^2])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx &=-\text {Subst}\left (\int \frac {1-12 x^2+16 x^4}{\left (9-4 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )\\ &=-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {1}{27} \text {Subst}\left (\int \frac {52+108 x^2}{\left (9-4 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )\\ &=-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}-\text {Subst}\left (\int \frac {1}{\sqrt {9-4 x^2}} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{2} \sin ^{-1}\left (\frac {2 \cos (x)}{3}\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 63, normalized size = 1.31 \begin {gather*} \frac {2550 \cos (x)-590 \cos (3 x)+243 i (7-2 \cos (2 x))^{3/2} \log \left (2 i \cos (x)+\sqrt {7-2 \cos (2 x)}\right )}{486 (7-2 \cos (2 x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]

[Out]

(2550*Cos[x] - 590*Cos[3*x] + (243*I)*(7 - 2*Cos[2*x])^(3/2)*Log[(2*I)*Cos[x] + Sqrt[7 - 2*Cos[2*x]]])/(486*(7
- 2*Cos[2*x])^(3/2))

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Maple [A]
time = 0.12, size = 53, normalized size = 1.10

method	result	size
derivativedivides	\(-\frac {4 \left (\cos ^{3}\left (x \right )\right )}{3 \left (9-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}+\frac {214 \cos \left (x \right )}{243 \sqrt {9-4 \left (\cos ^{2}\left (x \right )\right )}}-\frac {\arcsin \left (\frac {2 \cos \left (x \right )}{3}\right )}{2}+\frac {26 \cos \left (x \right )}{27 \left (9-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\)	\(53\)
default	\(-\frac {4 \left (\cos ^{3}\left (x \right )\right )}{3 \left (9-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}+\frac {214 \cos \left (x \right )}{243 \sqrt {9-4 \left (\cos ^{2}\left (x \right )\right )}}-\frac {\arcsin \left (\frac {2 \cos \left (x \right )}{3}\right )}{2}+\frac {26 \cos \left (x \right )}{27 \left (9-4 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-4/3*cos(x)^3/(9-4*cos(x)^2)^(3/2)+214/243*cos(x)/(9-4*cos(x)^2)^(1/2)-1/2*arcsin(2/3*cos(x))+26/27*cos(x)/(9-
4*cos(x)^2)^(3/2)

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Maxima [A]
time = 1.61, size = 69, normalized size = 1.44 \begin {gather*} -2 \, {\left (\frac {2 \, \cos \left (x\right )^{2}}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}}\right )} \cos \left (x\right ) + \frac {52 \, \cos \left (x\right )}{243 \, \sqrt {-4 \, \cos \left (x\right )^{2} + 9}} + \frac {26 \, \cos \left (x\right )}{27 \, {\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-2*(2*cos(x)^2/(-4*cos(x)^2 + 9)^(3/2) - 3/(-4*cos(x)^2 + 9)^(3/2))*cos(x) + 52/243*cos(x)/sqrt(-4*cos(x)^2 +
9) + 26/27*cos(x)/(-4*cos(x)^2 + 9)^(3/2) - 1/2*arcsin(2/3*cos(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (36) = 72\).
time = 1.30, size = 131, normalized size = 2.73 \begin {gather*} \frac {243 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (-\frac {81 \, \cos \left (x\right ) \sin \left (x\right ) - 4 \, {\left (8 \, \cos \left (x\right )^{3} - 9 \, \cos \left (x\right )\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9}}{64 \, \cos \left (x\right )^{4} - 225 \, \cos \left (x\right )^{2} + 81}\right ) - 243 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) - 80 \, {\left (59 \, \cos \left (x\right )^{3} - 108 \, \cos \left (x\right )\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9}}{972 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/972*(243*(16*cos(x)^4 - 72*cos(x)^2 + 81)*arctan(-(81*cos(x)*sin(x) - 4*(8*cos(x)^3 - 9*cos(x))*sqrt(-4*cos(
x)^2 + 9))/(64*cos(x)^4 - 225*cos(x)^2 + 81)) - 243*(16*cos(x)^4 - 72*cos(x)^2 + 81)*arctan(sin(x)/cos(x)) - 8
0*(59*cos(x)^3 - 108*cos(x))*sqrt(-4*cos(x)^2 + 9))/(16*cos(x)^4 - 72*cos(x)^2 + 81)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)/(5*cos(x)**2+9*sin(x)**2)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 1.26, size = 40, normalized size = 0.83 \begin {gather*} -\frac {20 \, {\left (59 \, \cos \left (x\right )^{2} - 108\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9} \cos \left (x\right )}{243 \, {\left (4 \, \cos \left (x\right )^{2} - 9\right )}^{2}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x, algorithm="giac")

[Out]

-20/243*(59*cos(x)^2 - 108)*sqrt(-4*cos(x)^2 + 9)*cos(x)/(4*cos(x)^2 - 9)^2 - 1/2*arcsin(2/3*cos(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sin \left (5\,x\right )}{{\left (5\,{\cos \left (x\right )}^2+9\,{\sin \left (x\right )}^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x)

[Out]

int(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2), x)

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