3.1.7 \(\int \frac {x^2}{a+b \cos ^2(x)} \, dx\) [7]

Optimal. Leaf size=311 \[ -\frac {i x^2 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^2 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {x \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {i \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {i \text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}} \]

[Out]

-1/2*I*x^2*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^2*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*x*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)+1/2*x*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)-1/4*I*pol
ylog(3,-b*exp(2*I*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)+1/4*I*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)

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4682, 3402, 2296, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {x \text {Li}_2\left (-\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {x \text {Li}_2\left (-\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {i \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 {i \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 {i x^2 \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^2 \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^2/(a + b*Cos[x]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.21, size = 239, normalized size = 0.77 \begin {gather*} -\frac {i \left (2 x^2 \log \left (1+\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )-2 x^2 \log \left (1+\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )-2 i x \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )+2 i x \text {PolyLog}\left (2,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )+\text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )-\text {PolyLog}\left (3,-\frac {b e^{2 i x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )\right )}{4 \sqrt {a} \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Cos[x]^2),x]

[Out]

((-1/4*I)*(2*x^2*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 2*x^2*Log[1 + (b*E^((2*I)*x))/(2
*a + b + 2*Sqrt[a]*Sqrt[a + b])] - (2*I)*x*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] +
(2*I)*x*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] + PolyLog[3, -((b*E^((2*I)*x))/(2*a +
b - 2*Sqrt[a]*Sqrt[a + b]))] - PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]))/(Sqrt[a]*Sq
rt[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. 709 vs. \(2 (235 ) = 470\).
time = 0.09, size = 710, normalized size = 2.28

method	result	size
risch	\(-\frac {2 x^{3}}{3 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i \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 {b x \polylog \left (2, \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 )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i a \polylog \left (3, \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 )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {b \,x^{3}}{3 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i b \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 )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {2 a \,x^{3}}{3 \sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i b \,x^{2} \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 )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {x \polylog \left (2, \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 {i x^{2} \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 )}-2 a -b}-\frac {x \polylog \left (2, \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 )}-2 a -b}-\frac {i x^{2} \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 {a x \polylog \left (2, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{\sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {i \polylog \left (3, \frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{2 \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}-\frac {x^{3}}{3 \sqrt {a \left (a +b \right )}}-\frac {i a \,x^{2} \ln \left (1-\frac {b \,{\mathrm e}^{2 i x}}{-2 \sqrt {a \left (a +b \right )}-2 a -b}\right )}{\sqrt {a \left (a +b \right )}\, \left (-2 \sqrt {a \left (a +b \right )}-2 a -b \right )}\)	\(710\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(-2*(a*(a+b))^(1/2)-2*a-b)*x^3-1/4*I/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*polylog(3,b*exp(2*I*x)/
(-2*(a*(a+b))^(1/2)-2*a-b))-1/2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*x*polylog(2,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)*a*polylog(3,b*exp(2*I*x)/(-2*(a*(a+b))^(1
/2)-2*a-b))-1/3/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*x^3-1/4*I/(a*(a+b))^(1/2)*polylog(3,b*exp(2*I*x)/
(2*(a*(a+b))^(1/2)-2*a-b))-2/3/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*a*x^3-1/2*I/(a*(a+b))^(1/2)/(-2*(a*(
a+b))^(1/2)-2*a-b)*b*x^2*ln(1-b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))-1/2/(a*(a+b))^(1/2)*x*polylog(2,b*exp(2
*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))-I/(-2*(a*(a+b))^(1/2)-2*a-b)*x^2*ln(1-b*exp(2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b)
)-1/(-2*(a*(a+b))^(1/2)-2*a-b)*x*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^2*
ln(1-b*exp(2*I*x)/(2*(a*(a+b))^(1/2)-2*a-b))-1/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*a*x*polylog(2,b*exp(
2*I*x)/(-2*(a*(a+b))^(1/2)-2*a-b))-1/2*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))-1/3/(a*(a+b))^(1/2)*x^3-I/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*a*x^2*ln(1-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^2/(a+b*cos(x)^2),x, algorithm="maxima")

[Out]

integrate(x^2/(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. 2400 vs. \(2 (228) = 456\).
time = 2.72, size = 2400, normalized size = 7.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-I*b*x^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*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) - 2*b*x*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) - 2*b*x*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) - 2*b*x*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) - 2*b*x*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) + 2*b*x*sqrt((a^2 + a*b)/b^2)*dilog((((2*a + b)*cos(x) + (2*I*a + I*b)*si
n(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) + 2*b*x*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) + 2*b*x*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) + 2*b*x*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) - 2*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -((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) + 2*I*b*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) + 2*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -((2*a + b)*c
os(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) - 2*I*b*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) + 2*I*b*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) - 2*I*b*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) - 2*I*b*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) + 2*I*b*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))/(a^2 + a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^2/(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^2}{b\,{\cos \left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*cos(x)^2),x)

[Out]

int(x^2/(a + b*cos(x)^2), x)

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