Integrand size = 22, antiderivative size = 48 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=-\frac {1}{2} \arcsin \left (\frac {2 \cos (x)}{3}\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}} \]
-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)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.31 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=\frac {2550 \cos (x)-590 \cos (3 x)+243 i (7-2 \cos (2 x))^{3/2} \log \left (2 i \cos (x)+\sqrt {7-2 \cos (2 x)}\right )}{486 (7-2 \cos (2 x))^{3/2}} \]
(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))
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 4879, 1471, 27, 298, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin (5 x)}{\left (9 \sin ^2(x)+5 \cos ^2(x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (5 x)}{\left (9 \sin (x)^2+5 \cos (x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle -\int \frac {16 \cos ^4(x)-12 \cos ^2(x)+1}{\left (9-4 \cos ^2(x)\right )^{5/2}}d\cos (x)\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle \frac {1}{27} \int \frac {4 \left (27 \cos ^2(x)+13\right )}{\left (9-4 \cos ^2(x)\right )^{3/2}}d\cos (x)-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{27} \int \frac {27 \cos ^2(x)+13}{\left (9-4 \cos ^2(x)\right )^{3/2}}d\cos (x)-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {4}{27} \left (\frac {295 \cos (x)}{36 \sqrt {9-4 \cos ^2(x)}}-\frac {27}{4} \int \frac {1}{\sqrt {9-4 \cos ^2(x)}}d\cos (x)\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {4}{27} \left (\frac {295 \cos (x)}{36 \sqrt {9-4 \cos ^2(x)}}-\frac {27}{8} \arcsin \left (\frac {2 \cos (x)}{3}\right )\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}\) |
(-55*Cos[x])/(27*(9 - 4*Cos[x]^2)^(3/2)) + (4*((-27*ArcSin[(2*Cos[x])/3])/ 8 + (295*Cos[x])/(36*Sqrt[9 - 4*Cos[x]^2])))/27
3.5.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(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] + Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Cos[v], x]}, -d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Cos[v]/d , u/Sin[v], x], x], x, Cos[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[Nonfree Factors[Cos[v], x], u/Sin[v], x]]
Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(36)=72\).
Time = 0.49 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.19
method | result | size |
derivativedivides | \(-\frac {\arcsin \left (\frac {2 \cos \left (x \right )}{3}\right )}{2}-\frac {55 \sqrt {-4 \left (\cos \left (x \right )-\frac {3}{2}\right )^{2}-12 \cos \left (x \right )+18}}{2592 \left (\cos \left (x \right )-\frac {3}{2}\right )^{2}}-\frac {295 \sqrt {-\left (\cos \left (x \right )-\frac {3}{2}\right )^{2}-3 \cos \left (x \right )+\frac {9}{2}}}{972 \left (\cos \left (x \right )-\frac {3}{2}\right )}+\frac {55 \sqrt {-4 \left (\cos \left (x \right )+\frac {3}{2}\right )^{2}+12 \cos \left (x \right )+18}}{2592 \left (\cos \left (x \right )+\frac {3}{2}\right )^{2}}-\frac {295 \sqrt {-\left (\cos \left (x \right )+\frac {3}{2}\right )^{2}+3 \cos \left (x \right )+\frac {9}{2}}}{972 \left (\cos \left (x \right )+\frac {3}{2}\right )}\) | \(105\) |
default | \(-\frac {\arcsin \left (\frac {2 \cos \left (x \right )}{3}\right )}{2}-\frac {55 \sqrt {-4 \left (\cos \left (x \right )-\frac {3}{2}\right )^{2}-12 \cos \left (x \right )+18}}{2592 \left (\cos \left (x \right )-\frac {3}{2}\right )^{2}}-\frac {295 \sqrt {-\left (\cos \left (x \right )-\frac {3}{2}\right )^{2}-3 \cos \left (x \right )+\frac {9}{2}}}{972 \left (\cos \left (x \right )-\frac {3}{2}\right )}+\frac {55 \sqrt {-4 \left (\cos \left (x \right )+\frac {3}{2}\right )^{2}+12 \cos \left (x \right )+18}}{2592 \left (\cos \left (x \right )+\frac {3}{2}\right )^{2}}-\frac {295 \sqrt {-\left (\cos \left (x \right )+\frac {3}{2}\right )^{2}+3 \cos \left (x \right )+\frac {9}{2}}}{972 \left (\cos \left (x \right )+\frac {3}{2}\right )}\) | \(105\) |
-1/2*arcsin(2/3*cos(x))-55/2592/(cos(x)-3/2)^2*(-4*(cos(x)-3/2)^2-12*cos(x )+18)^(1/2)-295/972/(cos(x)-3/2)*(-(cos(x)-3/2)^2-3*cos(x)+9/2)^(1/2)+55/2 592/(cos(x)+3/2)^2*(-4*(cos(x)+3/2)^2+12*cos(x)+18)^(1/2)-295/972/(cos(x)+ 3/2)*(-(cos(x)+3/2)^2+3*cos(x)+9/2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.73 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=\frac {243 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (-\frac {81 \, \cos \left (x\right ) \sin \left (x\right ) - 4 \, {\left (8 \, \cos \left (x\right )^{3} - 9 \, \cos \left (x\right )\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9}}{64 \, \cos \left (x\right )^{4} - 225 \, \cos \left (x\right )^{2} + 81}\right ) - 243 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) - 80 \, {\left (59 \, \cos \left (x\right )^{3} - 108 \, \cos \left (x\right )\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9}}{972 \, {\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )}} \]
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)) - 80* (59*cos(x)^3 - 108*cos(x))*sqrt(-4*cos(x)^2 + 9))/(16*cos(x)^4 - 72*cos(x) ^2 + 81)
Timed out. \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=-2 \, {\left (\frac {2 \, \cos \left (x\right )^{2}}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}}\right )} \cos \left (x\right ) + \frac {52 \, \cos \left (x\right )}{243 \, \sqrt {-4 \, \cos \left (x\right )^{2} + 9}} + \frac {26 \, \cos \left (x\right )}{27 \, {\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac {3}{2}}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \left (x\right )\right ) \]
-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))
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=-\frac {20 \, {\left (59 \, \cos \left (x\right )^{2} - 108\right )} \sqrt {-4 \, \cos \left (x\right )^{2} + 9} \cos \left (x\right )}{243 \, {\left (4 \, \cos \left (x\right )^{2} - 9\right )}^{2}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \left (x\right )\right ) \]
-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))
Timed out. \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx=\int \frac {\sin \left (5\,x\right )}{{\left (5\,{\cos \left (x\right )}^2+9\,{\sin \left (x\right )}^2\right )}^{5/2}} \,d x \]