\(\int \frac {x^3}{a+b \cos ^2(x)} \, dx\) [8]
Optimal result
Integrand size = 14, antiderivative size = 415 \[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}
\]
[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)
Rubi [A] (verified)
Time = 0.72 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.00, number of
steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4682, 3402, 2296, 2221, 2611,
6744, 2320, 6724} \[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1+\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}}
\]
[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*}
\text {integral}& = 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 \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 \int x \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \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 i) \int \operatorname {PolyLog}\left (3,-\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 \operatorname {PolyLog}\left (3,-\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 \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 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\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 {\operatorname {PolyLog}\left (3,-\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 \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}} \\
\end{align*}
Mathematica [A] (verified)
Time = 4.40 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.70
\[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\frac {-4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+4 i x^3 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )-6 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+6 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )-6 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )+6 i x \operatorname {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )+3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a (a+b)}}\right )-3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a (a+b)}}\right )}{8 \sqrt {a (a+b)}}
\]
[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*(a + b)])] + (4*I)*x^3*Log[1 + (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*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*Sq
rt[a*(a + b)])
Maple [B] (verified)
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
| | |
method | result | size |
| | |
risch |
\(\text {Expression too large to display}\) |
\(915\) |
| | |
|
|
|
[In]
int(x^3/(a+b*cos(x)^2),x,method=_RETURNVERBOSE)
[Out]
-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)*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-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))^(1/2)*x^3*ln(1-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))
Fricas [B] (verification not implemented)
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
\[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\text {Too large to display}
\]
[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*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))*sqr
t(-(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*si
n(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)*poly
log(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)
)/(a^2 + a*b)
Sympy [F]
\[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int \frac {x^{3}}{a + b \cos ^{2}{\left (x \right )}}\, dx
\]
[In]
integrate(x**3/(a+b*cos(x)**2),x)
[Out]
Integral(x**3/(a + b*cos(x)**2), x)
Maxima [F]
\[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int { \frac {x^{3}}{b \cos \left (x\right )^{2} + a} \,d x }
\]
[In]
integrate(x^3/(a+b*cos(x)^2),x, algorithm="maxima")
[Out]
integrate(x^3/(b*cos(x)^2 + a), x)
Giac [F]
\[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int { \frac {x^{3}}{b \cos \left (x\right )^{2} + a} \,d x }
\]
[In]
integrate(x^3/(a+b*cos(x)^2),x, algorithm="giac")
[Out]
integrate(x^3/(b*cos(x)^2 + a), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {x^3}{a+b \cos ^2(x)} \, dx=\int \frac {x^3}{b\,{\cos \left (x\right )}^2+a} \,d x
\]
[In]
int(x^3/(a + b*cos(x)^2),x)
[Out]
int(x^3/(a + b*cos(x)^2), x)