3.1.8 \(\int \frac {x^3}{a+b \cos ^2(x)} \, dx\) [8]

Optimal. Leaf size=415 \[ -\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}} \]

[Out]

-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)

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Rubi [A]
time = 0.41, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4682, 3402, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \text {Li}_3\left (-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \text {Li}_3\left (-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {Li}_4\left (-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+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 \sqrt {a} \sqrt {a+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 \sqrt {a} \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Cos[x]^2),x]

[Out]

((-1/2*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) + ((I/2)*x^3*L
og[1 + (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) - (3*x^2*PolyLog[2, -((b*E^((
2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2
*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) - (((3*I)/4)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a +
b - 2*Sqrt[a]*Sqrt[a + b]))])/(Sqrt[a]*Sqrt[a + b]) + (((3*I)/4)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*S
qrt[a]*Sqrt[a + b]))])/(Sqrt[a]*Sqrt[a + b]) + (3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b
]))])/(8*Sqrt[a]*Sqrt[a + b]) - (3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a
]*Sqrt[a + b])

Rule 2221

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]
- Dist[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]

Rule 2296

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]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[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]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

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] + Dist[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]

Rule 3402

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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]

Rule 4682

Int[(Cos[(c_.) + (d_.)*(x_)]^2*(b_.) + (a_))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/2^n, Int[x^m*(2*a + b + b*Co
s[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]))

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 6744

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] - Dist[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]

Rubi steps

\begin {align*} \int \frac {x^3}{a+b \cos ^2(x)} \, dx &=2 \int \frac {x^3}{2 a+b+b \cos (2 x)} \, dx\\ &=4 \int \frac {e^{2 i x} x^3}{b+2 (2 a+b) e^{2 i x}+b e^{4 i x}} \, dx\\ &=\frac {(2 b) \int \frac {e^{2 i x} x^3}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{2 i x}} \, dx}{\sqrt {a} \sqrt {a+b}}-\frac {(2 b) \int \frac {e^{2 i x} x^3}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{2 i x}} \, dx}{\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 {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 i) \int x^2 \log \left (1+\frac {2 b e^{2 i x}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt {a} \sqrt {a+b}}-\frac {(3 i) \int x^2 \log \left (1+\frac {2 b e^{2 i x}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{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 {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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \int x \text {Li}_2\left (-\frac {2 b e^{2 i x}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt {a} \sqrt {a+b}}-\frac {3 \int x \text {Li}_2\left (-\frac {2 b e^{2 i x}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{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 {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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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) \int \text {Li}_3\left (-\frac {2 b e^{2 i x}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 \sqrt {a} \sqrt {a+b}}-\frac {(3 i) \int \text {Li}_3\left (-\frac {2 b e^{2 i x}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 \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 {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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{x} \, dx,x,e^{2 i x}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{x} \, dx,x,e^{2 i x}\right )}{8 \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 {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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {Li}_4\left (-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Li}_4\left (-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}\\ \end {align*}

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Mathematica [A]
time = 1.13, size = 319, normalized size = 0.77 \begin {gather*} \frac {-4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )+4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )-6 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )+6 x^2 \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )-6 i x \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )+6 i x \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )+3 \text {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )-3 \text {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Cos[x]^2),x]

[Out]

((-4*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (4*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*
a + b + 2*Sqrt[a]*Sqrt[a + b])] - 6*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + 6*x
^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] - (6*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*
a + b - 2*Sqrt[a]*Sqrt[a + b]))] + (6*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] +
3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] - 3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b +
2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b])

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Maple [B] 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 = 0.10, size = 915, normalized size = 2.20

method	result	size
risch	\(-\frac {3 i \polylog \left (3, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 i x \polylog \left (3, \frac {b \,{\mathrm e}^{2 i x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{4 \sqrt {a \left (a +b \right )}}-\frac {i x^{3} \ln \left (1-\frac {b \,{\mathrm e}^{2 i x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{2 \sqrt {a \left (a +b \right )}}-\frac {x^{4}}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {a \,x^{4}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {b \,x^{4}}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 i \polylog \left (3, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a x}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 i \polylog \left (3, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b x}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i \ln \left (1-\frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x^{3}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}-\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) x^{2}}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a \,x^{2}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 \polylog \left (2, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b \,x^{2}}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{4 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a}{4 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b}{8 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i \ln \left (1-\frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) a \,x^{3}}{\sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {x^{4}}{4 \sqrt {a \left (a +b \right )}}-\frac {i \ln \left (1-\frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right ) b \,x^{3}}{2 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {3 x^{2} \polylog \left (2, \frac {b \,{\mathrm e}^{2 i x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{4 \sqrt {a \left (a +b \right )}}+\frac {3 \polylog \left (4, \frac {b \,{\mathrm e}^{2 i x}}{2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{8 \sqrt {a \left (a +b \right )}}\)	\(915\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

-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)*x^3*ln(1-b*ex
p(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))-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-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^4-3/2*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-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-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)*polylog(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)*polylog(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-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^3-1/4/(a*(a+b))^(1/2)*x^4-3/4*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-3/4/(a*(a+b))^(1/2)*x^2*polylog(2,b*exp(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))+3/8/(a*(
a+b))^(1/2)*polylog(4,b*exp(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*cos(x)^2),x, algorithm="maxima")

[Out]

integrate(x^3/(b*cos(x)^2 + a), x)

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Fricas [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 = 3.10, size = 3188, normalized size = 7.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*cos(x)^2),x, algorithm="fricas")

[Out]

-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*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))*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*s
in(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*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))*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))*sqr
t((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + b)/b) - 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog((((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 + 1) - 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cos(x) - (2*I*a + I*b)*sin(x) - 2*(b*c
os(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 + 1) - 3*b*x^
2*sqrt((a^2 + a*b)/b^2)*dilog((((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 + 1) - 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog
(-(((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 + 1) + 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog((((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 + 1) + 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-(((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 + 1) + 3*b*x^2*sqrt((
a^2 + a*b)/b^2)*dilog((((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 + 1) + 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-(((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 + 1) - 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -((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)
+ 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, ((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) + 6*I*b*x*sqrt((a^2 + a*b)/b^2)*po
lylog(3, -((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) - 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, ((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) + 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -((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) - 6*I*b*x*sqrt((a^2 + a*b)
/b^2)*polylog(3, ((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))*s
qrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -((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) + 6*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, ((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) + 6*b*sqrt((a^2 + a*b
)/b^2)*polylog(4, -((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) + 6*b*sqrt((a^2 + a*b)/b^2)*polylog(4, ((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) + 6*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -((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*s...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*cos(x)**2),x)

[Out]

Integral(x**3/(a + b*cos(x)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*cos(x)^2),x, algorithm="giac")

[Out]

integrate(x^3/(b*cos(x)^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{b\,{\cos \left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*cos(x)^2),x)

[Out]

int(x^3/(a + b*cos(x)^2), x)

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