3.579 \(\int \frac{x^3}{a+b \cos (x) \sin (x)} \, dx\)

Optimal. Leaf size=461 \[ -\frac{3 x^2 \text{PolyLog}\left (2,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 x^2 \text{PolyLog}\left (2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{2 \sqrt{4 a^2-b^2}}-\frac{3 i x \text{PolyLog}\left (3,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 i x \text{PolyLog}\left (3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 \text{PolyLog}\left (4,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{4 \sqrt{4 a^2-b^2}}-\frac{3 \text{PolyLog}\left (4,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{4 \sqrt{4 a^2-b^2}}-\frac{i x^3 \log \left (1-\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{\sqrt{4 a^2-b^2}}+\frac{i x^3 \log \left (1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{\sqrt{4 a^2-b^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.629673, antiderivative size = 461, normalized size of antiderivative = 1., 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 = {4584, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3 x^2 \text{PolyLog}\left (2,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 x^2 \text{PolyLog}\left (2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{2 \sqrt{4 a^2-b^2}}-\frac{3 i x \text{PolyLog}\left (3,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 i x \text{PolyLog}\left (3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{2 \sqrt{4 a^2-b^2}}+\frac{3 \text{PolyLog}\left (4,\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{4 \sqrt{4 a^2-b^2}}-\frac{3 \text{PolyLog}\left (4,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{4 \sqrt{4 a^2-b^2}}-\frac{i x^3 \log \left (1-\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right )}{\sqrt{4 a^2-b^2}}+\frac{i x^3 \log \left (1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{\sqrt{4 a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Cos[x]*Sin[x]),x]

[Out]

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

Rule 4584

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cos[(c_.) + (d_.)*(x_)]*(b_.)*Sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol]
:> Int[(e + f*x)^m*(a + (b*Sin[2*c + 2*d*x])/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 6609

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]

Rule 2282

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 6589

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

Mathematica [A]  time = 0.863064, size = 340, normalized size = 0.74 \[ \frac{-6 x^2 \text{PolyLog}\left (2,-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right )+6 x^2 \text{PolyLog}\left (2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )-6 i x \text{PolyLog}\left (3,-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right )+6 i x \text{PolyLog}\left (3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )+3 \text{PolyLog}\left (4,-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right )-3 \text{PolyLog}\left (4,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )-4 i x^3 \log \left (1+\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right )+4 i x^3 \log \left (1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right )}{4 \sqrt{4 a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Cos[x]*Sin[x]),x]

[Out]

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

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Maple [B]  time = 0.148, size = 2282, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*cos(x)*sin(x)),x)

[Out]

-2/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*b^2*x^
3-2/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*(-(2*a+b)*(2*a-b))^(1/2)*a*x^4+12/(8*a^2-2*b^2)/(-2*I*a-(-
(2*a+b)*(2*a-b))^(1/2))*polylog(3,b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*a^2*x-3/(8*a^2-2*b^2)/(-2*I*
a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(3,b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*b^2*x+3/(8*a^2-2*b^2)/(-
2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(2*a-b))^(
1/2)*a+6/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/
2)))*(-(2*a+b)*(2*a-b))^(1/2)*a*x^2-6/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(
-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(2*a-b))^(1/2)*a*x^2+I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1
/2))*b^2*x^4+I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*b^2*x^4-4*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*
a-b))^(1/2))*a^2*x^4+6*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a-(-(2*a
+b)*(2*a-b))^(1/2)))*a^2-3/2*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a-
(-(2*a+b)*(2*a-b))^(1/2)))*b^2-4*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*a^2*x^4+6*I/(8*a^2-2*b^2)/(
-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*a^2-3/2*I/(8*a^2-2*
b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*b^2-3/(8*a^2-
2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(4,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(
2*a-b))^(1/2)*a+8/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))
^(1/2)))*a^2*x^3-2/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b)
)^(1/2)))*b^2*x^3+2/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*(-(2*a+b)*(2*a-b))^(1/2)*a*x^4+12/(8*a^2-2
*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(3,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*a^2*x-3/(8*a
^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(3,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*b^2*x+8/
(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*a^2*x^3-4
*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a
+b)*(2*a-b))^(1/2)*a*x^3-6*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(3,b*exp(2*I*x)/(-2*I*a-(-
(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(2*a-b))^(1/2)*a*x+6*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*poly
log(3,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(2*a-b))^(1/2)*a*x+4*I/(8*a^2-2*b^2)/(-2*I*a+(
-(2*a+b)*(2*a-b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*(-(2*a+b)*(2*a-b))^(1/2)*a*x^3-1
2*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2)))*
a^2*x^2+3*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a+b)*(2*a-b))^
(1/2)))*b^2*x^2-12*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a-(-(2*a+b)*
(2*a-b))^(1/2)))*a^2*x^2+3*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a+b)*(2*a-b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a-(-
(2*a+b)*(2*a-b))^(1/2)))*b^2*x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{b \cos \left (x\right ) \sin \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 5.08906, size = 7900, normalized size = 17.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2*((4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt(-(
4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b) + 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*l
og(1/2*((-4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2
- b^2)/b^2) + 2*I*a)/b) + 2*b)/b) - 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2*((4*I*a*cos(x) - 4*a*sin(x) - 2*
(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b) - 2*
b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2*((-4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 -
b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b) + 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2
*((4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)
/b^2) - 2*I*a)/b) + 2*b)/b) + 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2*((-4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos
(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) + 2*b)/b) - 2*b*x^3*s
qrt(-(4*a^2 - b^2)/b^2)*log(1/2*((4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^
2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) + 2*b)/b) - 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(1/2*((-4*I*
a*cos(x) - 4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2)
- 2*I*a)/b) + 2*b)/b) + 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*((4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x
) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b + 1) + 6*I*b*x
^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*((-4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 -
b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b + 1) + 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*di
log(-1/2*((4*I*a*cos(x) - 4*a*sin(x) - 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a
^2 - b^2)/b^2) + 2*I*a)/b) + 2*b)/b + 1) + 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*((-4*I*a*cos(x) + 4*a
*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) +
2*b)/b + 1) - 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*((4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) + I*b*si
n(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) + 2*b)/b + 1) - 6*I*b*x^2*sqrt(-(
4*a^2 - b^2)/b^2)*dilog(-1/2*((-4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2)
)*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) + 2*b)/b + 1) - 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*(
(4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/
b^2) - 2*I*a)/b) + 2*b)/b + 1) - 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(-1/2*((-4*I*a*cos(x) - 4*a*sin(x) -
2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) + 2*b)/b + 1
) + 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt
(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) + 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylo
g(3, 1/2*(-4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^
2 - b^2)/b^2) + 2*I*a)/b)/b) - 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(4*I*a*cos(x) - 4*a*sin(x) - 2*(
b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) - 12*b*x*sqr
t(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(-4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^
2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) + 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(4*
I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2)
- 2*I*a)/b)/b) + 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(-4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) + I
*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b)/b) - 12*b*x*sqrt(-(4*a^2 - b
^2)/b^2)*polylog(3, 1/2*(4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(
-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b)/b) - 12*b*x*sqrt(-(4*a^2 - b^2)/b^2)*polylog(3, 1/2*(-4*I*a*cos(x) -
4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b)
/b) - 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sq
rt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) - 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*poly
log(4, 1/2*(-4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*
a^2 - b^2)/b^2) + 2*I*a)/b)/b) - 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(4*I*a*cos(x) - 4*a*sin(x) - 2
*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) - 12*I*b*s
qrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(-4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 -
b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b)/b) + 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(
4*I*a*cos(x) - 4*a*sin(x) + 2*(b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^
2) - 2*I*a)/b)/b) + 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(-4*I*a*cos(x) + 4*a*sin(x) - 2*(b*cos(x) +
I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b)/b) + 12*I*b*sqrt(-(4*a^2 -
b^2)/b^2)*polylog(4, 1/2*(4*I*a*cos(x) + 4*a*sin(x) + 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqr
t(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b)/b) + 12*I*b*sqrt(-(4*a^2 - b^2)/b^2)*polylog(4, 1/2*(-4*I*a*cos(x)
- 4*a*sin(x) - 2*(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/
b)/b))/(4*a^2 - b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a + b*sin(x)*cos(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{b \cos \left (x\right ) \sin \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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