3.3.1 \(\int \frac {x^4 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^3} \, dx\) [201]

Optimal. Leaf size=343 \[ \frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b (a+b \text {ArcSin}(c x))}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b (a+b \text {ArcSin}(c x))}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 (a+b \text {ArcSin}(c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {ArcSin}(c x))^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \text {ArcSin}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac {3 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}-\frac {3 b^2 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}+\frac {3 b^2 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3} \]

[Out]

1/12*b^2*x/c^4/d^3/(-c^2*x^2+1)-1/6*b*(a+b*arcsin(c*x))/c^5/d^3/(-c^2*x^2+1)^(3/2)+1/4*x^3*(a+b*arcsin(c*x))^2
/c^2/d^3/(-c^2*x^2+1)^2-3/8*x*(a+b*arcsin(c*x))^2/c^4/d^3/(-c^2*x^2+1)-3/4*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+
(-c^2*x^2+1)^(1/2))/c^5/d^3-7/6*b^2*arctanh(c*x)/c^5/d^3+3/4*I*b*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x
^2+1)^(1/2)))/c^5/d^3-3/4*I*b*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3-3/4*b^2*polylo
g(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3+3/4*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3+5/4*b*(a+b*
arcsin(c*x))/c^5/d^3/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.38, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {4791, 4749, 4266, 2611, 2320, 6724, 4767, 212, 272, 45, 4779, 12, 393} \begin {gather*} -\frac {3 i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{4 c^5 d^3}+\frac {3 i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 c^5 d^3}-\frac {3 i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 c^5 d^3}+\frac {x^3 (a+b \text {ArcSin}(c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {5 b (a+b \text {ArcSin}(c x))}{4 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {b (a+b \text {ArcSin}(c x))}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 x (a+b \text {ArcSin}(c x))^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 b^2 \text {Li}_3\left (-i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}+\frac {3 b^2 \text {Li}_3\left (i e^{i \text {ArcSin}(c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

(b^2*x)/(12*c^4*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c^5*d^3*(1 - c^2*x^2)^(3/2)) + (5*b*(a + b*Arc
Sin[c*x]))/(4*c^5*d^3*Sqrt[1 - c^2*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a +
b*ArcSin[c*x])^2)/(8*c^4*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^
5*d^3) - (7*b^2*ArcTanh[c*x])/(6*c^5*d^3) + (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])
])/(c^5*d^3) - (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^3) - (3*b^2*PolyLog[3,
(-I)*E^(I*ArcSin[c*x])])/(4*c^5*d^3) + (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4779

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 4791

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p + 1
))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(2*c*(p + 1)))*Simp[(d
+ e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]

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^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}-\frac {3 \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{2 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {b^2 \int \frac {-2+3 c^2 x^2}{3 c^4 \left (1-c^2 x^2\right )^2} \, dx}{2 d^3}+\frac {(3 b) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 c^3 d^3}+\frac {3 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{8 c^4 d^2}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^5 d^3}+\frac {b^2 \int \frac {-2+3 c^2 x^2}{\left (1-c^2 x^2\right )^2} \, dx}{6 c^4 d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{4 c^4 d^3}\\ &=\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 b^2 \tanh ^{-1}(c x)}{4 c^5 d^3}-\frac {(3 b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}+\frac {(3 b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac {\left (5 b^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{12 c^4 d^3}\\ &=\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}+\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}\\ &=\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}\\ &=\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 b^2 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac {3 b^2 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1012\) vs. \(2(343)=686\).
time = 5.97, size = 1012, normalized size = 2.95 \begin {gather*} \frac {\frac {24 a^2 c x}{\left (-1+c^2 x^2\right )^2}+\frac {60 a^2 c x}{-1+c^2 x^2}-\frac {60 a b \left (\sqrt {1-c^2 x^2}-\text {ArcSin}(c x)\right )}{-1+c x}+\frac {60 a b \left (\sqrt {1-c^2 x^2}+\text {ArcSin}(c x)\right )}{1+c x}+\frac {4 a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \text {ArcSin}(c x)\right )}{(-1+c x)^2}-\frac {4 a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \text {ArcSin}(c x)\right )}{(1+c x)^2}-18 a^2 \log (1-c x)+18 a^2 \log (1+c x)+18 a b \left (i \text {ArcSin}(c x)^2+\text {ArcSin}(c x) \left (-3 i \pi -4 \log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+\log \left (1+i e^{i \text {ArcSin}(c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-\log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )\right )+4 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )\right )+18 a b \left (-i \text {ArcSin}(c x)^2+\text {ArcSin}(c x) \left (i \pi +4 \log \left (1-i e^{i \text {ArcSin}(c x)}\right )\right )+2 \pi \left (2 \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+\log \left (1-i e^{i \text {ArcSin}(c x)}\right )-2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-\log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )\right )-4 i \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )\right )+4 b^2 \left (9 \text {ArcSin}(c x)^2 \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+9 \pi \text {ArcSin}(c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (1-i e^{i \text {ArcSin}(c x)}\right )\right )-9 \text {ArcSin}(c x)^2 \log \left (1+i e^{i \text {ArcSin}(c x)}\right )-9 \text {ArcSin}(c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (-i+e^{i \text {ArcSin}(c x)}\right )\right )+9 \pi \text {ArcSin}(c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left (-i+e^{i \text {ArcSin}(c x)}\right )\right )+9 \text {ArcSin}(c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcSin}(c x)} \left ((1+i)+(1-i) e^{i \text {ArcSin}(c x)}\right )\right )-9 \pi \text {ArcSin}(c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+28 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+9 \text {ArcSin}(c x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-28 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-9 \text {ArcSin}(c x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-9 \pi \text {ArcSin}(c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )+18 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-18 i \text {ArcSin}(c x) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )-18 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )+18 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )\right )+\frac {b^2 \left (\text {ArcSin}(c x) \left (74 \sqrt {1-c^2 x^2}+30 \cos (3 \text {ArcSin}(c x))\right )+3 \text {ArcSin}(c x)^2 (3 c x-5 \sin (3 \text {ArcSin}(c x)))+2 (c x+\sin (3 \text {ArcSin}(c x)))\right )}{\left (-1+c^2 x^2\right )^2}}{96 c^5 d^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

((24*a^2*c*x)/(-1 + c^2*x^2)^2 + (60*a^2*c*x)/(-1 + c^2*x^2) - (60*a*b*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(-1
+ c*x) + (60*a*b*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(1 + c*x) + (4*a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSi
n[c*x]))/(-1 + c*x)^2 - (4*a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(1 + c*x)^2 - 18*a^2*Log[1 - c*x
] + 18*a^2*Log[1 + c*x] + 18*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-3*I)*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) +
2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] - Log[-Cos[
(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) + 18*a*b*((-I)*ArcSin[c*x]^2 + ArcSin[c*
x]*(I*Pi + 4*Log[1 - I*E^(I*ArcSin[c*x])]) + 2*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x
])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) +
4*b^2*(9*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 9*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x
])))/(2*E^((I/2)*ArcSin[c*x]))] - 9*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] - 9*ArcSin[c*x]^2*Log[((1/2 + I
/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 9*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin
[c*x])))/E^((I/2)*ArcSin[c*x])] + 9*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin
[c*x]))] - 9*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]
] + 9*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]
/2]] - 9*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 9*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[
c*x])/4]] + (18*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (18*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSi
n[c*x])] - 18*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 18*PolyLog[3, I*E^(I*ArcSin[c*x])]) + (b^2*(ArcSin[c*x]*(74
*Sqrt[1 - c^2*x^2] + 30*Cos[3*ArcSin[c*x]]) + 3*ArcSin[c*x]^2*(3*c*x - 5*Sin[3*ArcSin[c*x]]) + 2*(c*x + Sin[3*
ArcSin[c*x]])))/(-1 + c^2*x^2)^2)/(96*c^5*d^3)

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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. 843 vs. \(2 (350 ) = 700\).
time = 0.62, size = 844, normalized size = 2.46

method	result	size
derivativedivides	\(\frac {\frac {5 b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 a^{2} \ln \left (c x +1\right )}{16 d^{3}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} c x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {7 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d^{3}}-\frac {5 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 a b \arcsin \left (c x \right ) c^{3} x^{3}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {5 a b \,c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 a b \arcsin \left (c x \right ) c x}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {13 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {13 a b \sqrt {-c^{2} x^{2}+1}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b^{2} c^{3} x^{3}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} c x}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 a^{2} \ln \left (c x -1\right )}{16 d^{3}}-\frac {a^{2}}{16 d^{3} \left (c x +1\right )^{2}}+\frac {5 a^{2}}{16 d^{3} \left (c x +1\right )}+\frac {a^{2}}{16 d^{3} \left (c x -1\right )^{2}}+\frac {5 a^{2}}{16 d^{3} \left (c x -1\right )}}{c^{5}}\)	\(844\)
default	\(\frac {\frac {5 b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 a^{2} \ln \left (c x +1\right )}{16 d^{3}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 d^{3}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} c x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {7 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 d^{3}}-\frac {5 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 a b \arcsin \left (c x \right ) c^{3} x^{3}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {5 a b \,c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 a b \arcsin \left (c x \right ) c x}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {13 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {13 a b \sqrt {-c^{2} x^{2}+1}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b^{2} c^{3} x^{3}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} c x}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 a^{2} \ln \left (c x -1\right )}{16 d^{3}}-\frac {a^{2}}{16 d^{3} \left (c x +1\right )^{2}}+\frac {5 a^{2}}{16 d^{3} \left (c x +1\right )}+\frac {a^{2}}{16 d^{3} \left (c x -1\right )^{2}}+\frac {5 a^{2}}{16 d^{3} \left (c x -1\right )}}{c^{5}}\)	\(844\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^5*(5/8*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*c^3*x^3+3/16*a^2/d^3*ln(c*x+1)-3/8*b^2/d^3/(c^4*x^4-2*c
^2*x^2+1)*arcsin(c*x)^2*c*x-5/4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+5/4*a*b/d
^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*c^3*x^3-5/4*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*c^2*x^2*(-c^2*x^2+1)^(1/2)-3/4*
a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*c*x+13/12*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*(-c^2*x^2+1)^(1/
2)-3/4*I*b^2/d^3*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*I*b^2/d^3*arcsin(c*x)*polylog(2,-I*(I
*c*x+(-c^2*x^2+1)^(1/2)))+13/12*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(-c^2*x^2+1)^(1/2)-3/4*a*b/d^3*arcsin(c*x)*ln(1+
I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*a*b/d^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*I*a*b/d^3*dilog(1
+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*I*a*b/d^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-1/12*b^2/d^3/(c^4*x^4-2*c^2
*x^2+1)*c^3*x^3+1/12*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*c*x-3/16*a^2/d^3*ln(c*x-1)-1/16*a^2/d^3/(c*x+1)^2+5/16*a^2/
d^3/(c*x+1)+1/16*a^2/d^3/(c*x-1)^2+5/16*a^2/d^3/(c*x-1)+3/4*b^2/d^3*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/
4*b^2/d^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/8*b^2/d^3*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))
)-3/8*b^2/d^3*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7/3*I*b^2/d^3*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))

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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^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/16*a^2*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c*x -
1)/(c^5*d^3)) + 1/16*(3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(
c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) +
2*(5*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 16*(c^9*d^3*x^4 - 2*c^7*d^3*x^2 +
c^5*d^3)*integrate(-1/8*(16*a*b*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (3*(b^2*c^4*x^4 - 2*b^2*
c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)
*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) + 2*(5*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*
x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^10*d^3*x^6 - 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3)
, x))/(c^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5*d^3)

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Fricas [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^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x
^2 - d^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {a^{2} x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**3,x)

[Out]

-(Integral(a**2*x**4/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b**2*x**4*asin(c*x)**2/(c**6*x
**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x**4*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**
2 - 1), x))/d**3

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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^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate(-(b*arcsin(c*x) + a)^2*x^4/(c^2*d*x^2 - d)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3,x)

[Out]

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3, x)

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