Integrand size = 14, antiderivative size = 415 \[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}} \]
-1/2*I*x^3*ln(1+b*exp(2*I*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^ (1/2)+1/2*I*x^3*ln(1+b*exp(2*I*x)/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/( a+b)^(1/2)-3/4*x^2*polylog(2,-b*exp(2*I*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/ a^(1/2)/(a+b)^(1/2)+3/4*x^2*polylog(2,-b*exp(2*I*x)/(2*a+b+2*a^(1/2)*(a+b) ^(1/2)))/a^(1/2)/(a+b)^(1/2)-3/4*I*x*polylog(3,-b*exp(2*I*x)/(2*a+b-2*a^(1 /2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)+3/4*I*x*polylog(3,-b*exp(2*I*x)/(2*a +b+2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)+3/8*polylog(4,-b*exp(2*I*x) /(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)-3/8*polylog(4,-b*exp(2 *I*x)/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)
Time = 4.40 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.70 \[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\frac {-4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )-6 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+6 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )-6 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+6 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )+3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )-3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )}{8 \sqrt {a (a+b)}} \]
((-4*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a*(a + b)])] + (4*I) *x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a*(a + b)])] - 6*x^2*PolyLo g[2, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a*(a + b)]))] + 6*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a*(a + b)]))] - (6*I)*x*PolyLog[3, -(( b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a*(a + b)]))] + (6*I)*x*PolyLog[3, -((b*E ^((2*I)*x))/(2*a + b + 2*Sqrt[a*(a + b)]))] + 3*PolyLog[4, -((b*E^((2*I)*x ))/(2*a + b - 2*Sqrt[a*(a + b)]))] - 3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a*(a + b)]))])/(8*Sqrt[a*(a + b)])
Time = 1.30 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5097, 3042, 3802, 2694, 27, 2620, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {x^3}{a+b \cos ^2(x)} \, dx\) |
| \(\Big \downarrow \) 5097 |
| \(\displaystyle 2 \int \frac {x^3}{2 a+b+b \cos (2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \int \frac {x^3}{2 a+b+b \sin \left (2 x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3802 |
| \(\displaystyle 4 \int \frac {e^{2 i x} x^3}{e^{4 i x} b+b+2 (2 a+b) e^{2 i x}}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle 4 \left (\frac {b \int \frac {e^{2 i x} x^3}{2 \left (2 a-2 \sqrt {a+b} \sqrt {a}+b e^{2 i x}+b\right )}dx}{2 \sqrt {a} \sqrt {a+b}}-\frac {b \int \frac {e^{2 i x} x^3}{2 \left (2 a+2 \sqrt {a+b} \sqrt {a}+b e^{2 i x}+b\right )}dx}{2 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {b \int \frac {e^{2 i x} x^3}{2 a-2 \sqrt {a+b} \sqrt {a}+b e^{2 i x}+b}dx}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \int \frac {e^{2 i x} x^3}{2 a+2 \sqrt {a+b} \sqrt {a}+b e^{2 i x}+b}dx}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {3 i \int x^2 \log \left (\frac {e^{2 i x} b}{2 a-2 \sqrt {a+b} \sqrt {a}+b}+1\right )dx}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {3 i \int x^2 \log \left (\frac {e^{2 i x} b}{2 a+2 \sqrt {a+b} \sqrt {a}+b}+1\right )dx}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-i \int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )dx\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-i \int x \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )dx\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{2} i \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )dx-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{2} i \int \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )dx-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )de^{2 i x}-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )de^{2 i x}-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 4 \left (\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {3 i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-i \left (\frac {1}{4} \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-\frac {1}{2} i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )\right )\right )}{2 b}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\) |
4*((b*(((-1/2*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/b + (((3*I)/2)*((I/2)*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b - 2 *Sqrt[a]*Sqrt[a + b]))] - I*((-1/2*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + PolyLog[4, -((b*E^((2*I)*x))/(2*a + b - 2 *Sqrt[a]*Sqrt[a + b]))]/4)))/b))/(4*Sqrt[a]*Sqrt[a + b]) - (b*(((-1/2*I)*x ^3*Log[1 + (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/b + (((3*I) /2)*((I/2)*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] - I*((-1/2*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sq rt[a + b]))] + PolyLog[4, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/4)))/b))/(4*Sqrt[a]*Sqrt[a + b]))
3.1.8.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*( x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*E^(I*Pi*(k - 1/2))*(E^(I*(e + f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*( e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ [a^2 - b^2, 0] && IGtQ[m, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]^2*(b_.) + (a_))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1/2^n Int[x^m*(2*a + b + b*Cos[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || (EqQ[m, 1] && EqQ[n, -2]))
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (315 ) = 630\).
Time = 0.48 (sec) , antiderivative size = 915, normalized size of antiderivative = 2.20
-3/2*I/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(3,b*exp(2*I*x)/( -2*(a*(a+b))^(1/2)-2*a-b))*a*x-3/4*I/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2 *a-b)*polylog(3,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*b*x-3/2*I/(-2*(a* (a+b))^(1/2)-2*a-b)*polylog(3,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*x-1 /2/(-2*(a*(a+b))^(1/2)-2*a-b)*x^4-1/2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)- 2*a-b)*a*x^4-1/4/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*x^4-3/4*I/(a *(a+b))^(1/2)*x*polylog(3,b*exp(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))-1/2*I/(a *(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*x)/(-2*(a*(a+b))^( 1/2)-2*a-b))*b*x^3-I/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp (2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a*x^3-3/2/(-2*(a*(a+b))^(1/2)-2*a-b)*p olylog(2,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*x^2-3/2/(a*(a+b))^(1/2)/ (-2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a- b))*a*x^2-3/4/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2 *I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*b*x^2+3/4/(-2*(a*(a+b))^(1/2)-2*a-b)*pol ylog(4,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))+3/4/(a*(a+b))^(1/2)/(-2*(a *(a+b))^(1/2)-2*a-b)*polylog(4,b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a+ 3/8/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(4,b*exp(2*I*x)/(-2* (a*(a+b))^(1/2)-2*a-b))*b-I/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*x)/( -2*(a*(a+b))^(1/2)-2*a-b))*x^3-1/4/(a*(a+b))^(1/2)*x^4-3/4/(a*(a+b))^(1/2) *x^2*polylog(2,b*exp(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))-1/2*I/(a*(a+b))^...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3188 vs. \(2 (307) = 614\).
Time = 1.24 (sec) , antiderivative size = 3188, normalized size of antiderivative = 7.68 \[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\text {Too large to display} \]
-1/4*(-I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(x) + (2*I*a + I* b)*sin(x) - 2*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sq rt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b) + I*b*x^3*sqrt((a^2 + a*b)/b^2)* log((((2*a + b)*cos(x) - (2*I*a + I*b)*sin(x) - 2*(b*cos(x) - I*b*sin(x))* sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + b) /b) + I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(x) + (-2*I*a - I* b)*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sq rt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b) - I*b*x^3*sqrt((a^2 + a*b)/b^2)* log((((2*a + b)*cos(x) - (-2*I*a - I*b)*sin(x) - 2*(b*cos(x) + I*b*sin(x)) *sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + b )/b) + I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(x) + (2*I*a + I* b)*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqr t((a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b) - I*b*x^3*sqrt((a^2 + a*b)/b^2)*l og((((2*a + b)*cos(x) - (2*I*a + I*b)*sin(x) + 2*(b*cos(x) - I*b*sin(x))*s qrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + b)/b ) - I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(x) + (-2*I*a - I*b) *sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt( (a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b) + I*b*x^3*sqrt((a^2 + a*b)/b^2)*log ((((2*a + b)*cos(x) - (-2*I*a - I*b)*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sq rt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + b)...
\[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int \frac {x^{3}}{a + b \cos ^{2}{\left (x \right )}}\, dx \]
\[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int { \frac {x^{3}}{b \cos \left (x\right )^{2} + a} \,d x } \]
\[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int { \frac {x^{3}}{b \cos \left (x\right )^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int \frac {x^3}{b\,{\cos \left (x\right )}^2+a} \,d x \]