3.3 \(\int \frac {x^3}{a+b \sin ^2(x)} \, dx\)
Optimal. Leaf size=411 \[ -\frac {3 x^2 \text {Li}_2\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \text {Li}_2\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \text {Li}_3\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \text {Li}_3\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {Li}_4\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {Li}_4\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {i x^3 \log \left (1-\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1-\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}} \]
[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] time = 0.63, antiderivative size = 411, 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
= {4585, 3321, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac {3 x^2 \text {PolyLog}\left (2,\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \text {PolyLog}\left (2,\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 i x \text {PolyLog}\left (3,\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 i x \text {PolyLog}\left (3,\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \text {PolyLog}\left (4,\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \text {PolyLog}\left (4,\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {i x^3 \log \left (1-\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {i x^3 \log \left (1-\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{2 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
[In]
Int[x^3/(a + b*Sin[x]^2),x]
[Out]
((-I/2)*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*Log
[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*Sqrt[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 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 3321
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 4585
Int[(x_)^(m_.)*((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]^2)^(n_), 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 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 \sin ^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 \operatorname {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 \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 2.51, size = 315, normalized size = 0.77 \[ \frac {-6 x^2 \text {Li}_2\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )+6 x^2 \text {Li}_2\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-6 i x \text {Li}_3\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )+6 i x \text {Li}_3\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )+3 \text {Li}_4\left (\frac {b e^{2 i x}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )-3 \text {Li}_4\left (\frac {b e^{2 i x}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )-4 i x^3 \log \left (1-\frac {b e^{2 i x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )+4 i x^3 \log \left (1-\frac {b e^{2 i x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 \sqrt {a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3/(a + b*Sin[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])
________________________________________________________________________________________
fricas [C] time = 1.59, size = 3252, normalized size = 7.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(a+b*sin(x)^2),x, algorithm="fricas")
[Out]
1/16*(4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(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) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2
+ a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(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) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2
*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(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) + 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4
*I*a - 2*I*b)*sin(x) - 4*(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) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*
(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) + 2*b)/b) + 4*I*
b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(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) - 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/
b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(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) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a
+ b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(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) - 2*b)/b) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a
+ 2*I*b)*sin(x) - 4*(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) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*
(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) - 2*b)/b + 1) + 1
2*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(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) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2
+ a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(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) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-
1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(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) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*c
os(x) - (4*I*a + 2*I*b)*sin(x) + 4*(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) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a -
2*I*b)*sin(x) + 4*(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
) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*
(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) - 2*b)/b + 1) +
24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(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) - 24*I*b*x*sqrt((a^2 + a*b)/b
^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(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) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a
+ b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(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) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a -
2*I*b)*sin(x) - 4*(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) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(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) + 24*I*b*x*sqrt((a^2 +
a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(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) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2
*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*s
qrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (
-4*I*a - 2*I*b)*sin(x) + 4*(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) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(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) - 24*b*sqrt((a^2
+ a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(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) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*
(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(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) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a -
2*I*b)*sin(x) - 4*(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) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(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) + 24*b*sqrt((a^2 + a*b)/b^
2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(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) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b
)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(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) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*
sin(x) + 4*(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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \sin \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(a+b*sin(x)^2),x, algorithm="giac")
[Out]
integrate(x^3/(b*sin(x)^2 + a), x)
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maple [B] time = 0.14, size = 853, normalized size = 2.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(a+b*sin(x)^2),x)
[Out]
-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))-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))+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+3/8/(a*(a+b))^(1/2)*polylog(4,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/(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+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+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/(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*I/(a*(a+b))^(1/2)*x*polylog(3,b*exp(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+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^3-3/4/(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)*p
olylog(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*e
xp(2*I*x)/(2*(a*(a+b))^(1/2)+2*a+b))*b
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \sin \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(a+b*sin(x)^2),x, algorithm="maxima")
[Out]
integrate(x^3/(b*sin(x)^2 + a), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{b\,{\sin \relax (x)}^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(a + b*sin(x)^2),x)
[Out]
int(x^3/(a + b*sin(x)^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{a + b \sin ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/(a+b*sin(x)**2),x)
[Out]
Integral(x**3/(a + b*sin(x)**2), x)
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