3.579 \(\int \frac {x^3}{a+b \cos (x) \sin (x)} \, dx\)
Optimal. Leaf size=461 \[ -\frac {3 x^2 \text {Li}_2\left (\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {3 x^2 \text {Li}_2\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}-\frac {3 i x \text {Li}_3\left (\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {3 i x \text {Li}_3\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {3 \text {Li}_4\left (\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{4 \sqrt {4 a^2-b^2}}-\frac {3 \text {Li}_4\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{4 \sqrt {4 a^2-b^2}}-\frac {i x^3 \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x^3 \log \left (1-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}+2 a}\right )}{\sqrt {4 a^2-b^2}} \]
[Out]
-I*x^3*ln(1-I*b*exp(2*I*x)/(2*a-(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)+I*x^3*ln(1-I*b*exp(2*I*x)/(2*a+(4*a^2-b^
2)^(1/2)))/(4*a^2-b^2)^(1/2)-3/2*x^2*polylog(2,I*b*exp(2*I*x)/(2*a-(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)+3/2*x
^2*polylog(2,I*b*exp(2*I*x)/(2*a+(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)-3/2*I*x*polylog(3,I*b*exp(2*I*x)/(2*a-(
4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)+3/2*I*x*polylog(3,I*b*exp(2*I*x)/(2*a+(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/
2)+3/4*polylog(4,I*b*exp(2*I*x)/(2*a-(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)-3/4*polylog(4,I*b*exp(2*I*x)/(2*a+(
4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.63, antiderivative size = 461, 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
= {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 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 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 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 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 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 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 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]
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]
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 = 0.87, size = 340, normalized size = 0.74 \[ \frac {-6 x^2 \text {Li}_2\left (-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}-2 a}\right )+6 x^2 \text {Li}_2\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )-6 i x \text {Li}_3\left (-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}-2 a}\right )+6 i x \text {Li}_3\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )+3 \text {Li}_4\left (-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}-2 a}\right )-3 \text {Li}_4\left (\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )-4 i x^3 \log \left (1+\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}-2 a}\right )+4 i x^3 \log \left (1-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}+2 a}\right )}{4 \sqrt {4 a^2-b^2}} \]
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])
________________________________________________________________________________________
fricas [C] time = 1.29, size = 3324, normalized size = 7.21 \[ \text {result too large to display} \]
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(-((2*I*a*cos(x) - 2*a*sin(x) - (b*co
s(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)/b) - 2*b*x^3*s
qrt(-(4*a^2 - b^2)/b^2)*log(-((-2*I*a*cos(x) + 2*a*sin(x) + (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)/b) + 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(-((2*I*a*cos(x) -
2*a*sin(x) + (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)/b) + 2*b*x^3*sqrt(-(4*a^2 - b^2)/b^2)*log(-((-2*I*a*cos(x) + 2*a*sin(x) - (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)/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) + 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(((2*I*a*cos(x) - 2*a*sin
(x) - (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)/b +
1) + 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x) + 2*a*sin(x) + (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)/b + 1) - 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^
2)*dilog(((2*I*a*cos(x) - 2*a*sin(x) + (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)/b + 1) - 6*I*b*x^2*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x) + 2*a*sin(x) -
(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)/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*sqr
t(-(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)*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)*p
olylog(3, -(2*I*a*cos(x) - 2*a*sin(x) - (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, -(-2*I*a*cos(x) + 2*a*sin(x) + (b*co
s(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, -(2*I*a*cos(x) - 2*a*sin(x) + (b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*s
qrt((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, -(-2*I*a*cos(x) +
2*a*sin(x) - (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)*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*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*sqr
t(-(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, -(2*I*a*
cos(x) - 2*a*sin(x) - (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, -(-2*I*a*cos(x) + 2*a*sin(x) + (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, -(2*I*a*cos(x) - 2*a*sin(x) + (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, -(-2*I*a*cos(x) + 2*a*sin(x) - (b*co
s(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*c
os(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)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \cos \relax (x) \sin \relax (x) + a}\,{d x} \]
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)
________________________________________________________________________________________
maple [B] time = 0.30, size = 2282, normalized size = 4.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(a+b*cos(x)*sin(x)),x)
[Out]
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+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^2-1
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+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+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))*(-(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*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^3-6*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*polylog(3,b*e
xp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2)))*(-(2*a-b)*(2*a+b))^(1/2)*a*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^3-2/(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))*(-(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+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
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \cos \relax (x) \sin \relax (x) + a}\,{d x} \]
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)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{a+b\,\cos \relax (x)\,\sin \relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(a + b*cos(x)*sin(x)),x)
[Out]
int(x^3/(a + b*cos(x)*sin(x)), x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/(a+b*cos(x)*sin(x)),x)
[Out]
Timed out
________________________________________________________________________________________