3.3.9 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^4 (d-c^2 d x^2)^3} \, dx\) [209]

Optimal. Leaf size=572 \[ -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \text {ArcSin}(c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \text {ArcSin}(c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {ArcSin}(c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {ArcSin}(c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {ArcSin}(c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \text {ArcSin}(c x))^2 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \text {PolyLog}\left (3,i e^{i \text {ArcSin}(c x)}\right )}{4 d^3} \]

[Out]

-1/2*b^2*c^2/d^3/x+1/6*b^2*c^2/d^3/x/(-c^2*x^2+1)-1/12*b^2*c^4*x/d^3/(-c^2*x^2+1)+1/6*b*c^3*(a+b*arcsin(c*x))/
d^3/(-c^2*x^2+1)^(3/2)-1/3*b*c*(a+b*arcsin(c*x))/d^3/x^2/(-c^2*x^2+1)^(3/2)-1/3*(a+b*arcsin(c*x))^2/d^3/x^3/(-
c^2*x^2+1)^2-7/3*c^2*(a+b*arcsin(c*x))^2/d^3/x/(-c^2*x^2+1)^2+35/12*c^4*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)
^2+35/8*c^4*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)+35/4*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^
2+1)^(1/2)))/d^3-38/3*b*c^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d^3+17/6*b^2*c^3*arctanh(c*x)/
d^3+19/3*I*b^2*c^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d^3-19/3*I*b^2*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))
/d^3-35/4*I*c^3*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^3-35/4*I*b*c^3*(a+b*arcsin(c*x))*polylo
g(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3-35/4*b^2*c^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3+35/4*b^2*c^3*p
olylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3-29/12*b*c^3*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.88, antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 17, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 205, 4793, 4803, 4268, 2317, 2438, 296, 331} \begin {gather*} -\frac {35 i c^3 \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{4 d^3}+\frac {35 i b c^3 \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 d^3}-\frac {35 i b c^3 \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 d^3}-\frac {38 b c^3 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d^3}-\frac {7 c^2 (a+b \text {ArcSin}(c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {b c (a+b \text {ArcSin}(c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {ArcSin}(c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x (a+b \text {ArcSin}(c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {29 b c^3 (a+b \text {ArcSin}(c x))}{12 d^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 i b^2 c^3 \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{3 d^3}-\frac {19 i b^2 c^3 \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \text {Li}_3\left (-i e^{i \text {ArcSin}(c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \text {Li}_3\left (i e^{i \text {ArcSin}(c x)}\right )}{4 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^2}{2 d^3 x}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b^2*c^2)/(d^3*x) + (b^2*c^2)/(6*d^3*x*(1 - c^2*x^2)) - (b^2*c^4*x)/(12*d^3*(1 - c^2*x^2)) + (b*c^3*(a +
b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (b*c*(a + b*ArcSin[c*x]))/(3*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (29*
b*c^3*(a + b*ArcSin[c*x]))/(12*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7
*c^2*(a + b*ArcSin[c*x])^2)/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(12*d^3*(1 - c^2*x^2)
^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) - (((35*I)/4)*c^3*(a + b*ArcSin[c*x])^2*ArcTan[E^
(I*ArcSin[c*x])])/d^3 - (38*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d^3) + (17*b^2*c^3*ArcTan
h[c*x])/(6*d^3) + (((19*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d^3 + (((35*I)/4)*b*c^3*(a + b*ArcSin[c*
x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((35*I)/4)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c
*x])])/d^3 - (((19*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d^3 - (35*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[
c*x])])/(4*d^3) + (35*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*d^3)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 4268

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

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(p
+ 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a +
b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(
p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

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 4789

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4793

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(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]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

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 {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (7 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac {(2 b c) \int \frac {a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{3 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}+\frac {19 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{9 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{6 d^3}+\frac {\left (35 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {19 b^2 c^4 x}{18 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (5 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{18 d^3}+\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 d^3}-\frac {\left (7 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{9 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{18 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac {\left (35 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {62 b^2 c^3 \tanh ^{-1}(c x)}{9 d^3}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}+\frac {\left (14 b c^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{36 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{4 d^3}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {38 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}-\frac {\left (35 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d^3}+\frac {\left (35 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d^3}-\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}+\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}-\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}+\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^3}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {38 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}+\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {\left (14 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac {\left (14 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac {\left (35 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d^3}+\frac {\left (35 i b^2 c^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d^3}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {38 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac {\left (35 b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 d^3}+\frac {\left (35 b^2 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 d^3}\\ &=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {38 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d^3}+\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {35 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 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. \(1657\) vs. \(2(572)=1144\).
time = 10.18, size = 1657, normalized size = 2.90 \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*a^2/(d^3*x^3) - (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(-1 + c^2*x^2)^2) - (11*a^2*c^4*x)/(8*d^3*(-1 +
c^2*x^2)) - (35*a^2*c^3*Log[1 - c*x])/(16*d^3) + (35*a^2*c^3*Log[1 + c*x])/(16*d^3) - (2*a*b*((c^3*((2 - c*x)*
Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]))/(48*(-1 + c*x)^2) - (11*c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(16*(-1 + c
*x)) + (11*c^4*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(16*(c + c^2*x)) + (c^3*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*Arc
Sin[c*x]))/(48*(1 + c*x)^2) - 3*c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]) + (c*x*Sqrt[1 - c^2*x^2]
+ 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*x^3) + (35*c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2
)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[
c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]
])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/16 - (35*c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c
*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1
- I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*
I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/16))/d^3 - (b^2*c^3*(((-19*I)/3)*PolyLog[2, -E^(I*ArcSin[c*x])] + ((19
*I)/3)*PolyLog[2, E^(I*ArcSin[c*x])] + (68*ArcSin[c*x] + 35*ArcSin[c*x]^3 - 105*ArcSin[c*x]^2*Log[1 - I*E^(I*A
rcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] + 105*
ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 105*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((
I/2)*ArcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])]
- 105*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] + 105*Pi*ArcSin[c*x]*
Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 68*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 105*ArcSin[c*x]^2*Log[Co
s[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 68*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*ArcSin[c*x]^2*L
og[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (210*I)*Ar
cSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 210*PolyL
og[3, (-I)*E^(I*ArcSin[c*x])] - 210*PolyLog[3, I*E^(I*ArcSin[c*x])])/24 + (24 - 204*c*x*ArcSin[c*x] + 204*ArcS
in[c*x]^2 - 105*c*x*ArcSin[c*x]^3 + (20 + 658*ArcSin[c*x]^2)*Cos[2*ArcSin[c*x]] - 4*(6 + 35*ArcSin[c*x]^2)*Cos
[4*ArcSin[c*x]] - 20*Cos[6*ArcSin[c*x]] - 210*ArcSin[c*x]^2*Cos[6*ArcSin[c*x]] - 456*c*x*ArcSin[c*x]*Log[1 - E
^(I*ArcSin[c*x])] + 456*c*x*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + 540*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 204*
ArcSin[c*x]*Sin[3*ArcSin[c*x]] - 105*ArcSin[c*x]^3*Sin[3*ArcSin[c*x]] - 456*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*
x])]*Sin[3*ArcSin[c*x]] + 456*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] + 32*ArcSin[c*x]*Sin[4
*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[5*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[5*ArcSin[c*x]] + 152*ArcSin[c*x]*Log[
1 - E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 11
6*ArcSin[c*x]*Sin[6*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[7*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[7*ArcSin[c*x]] + 1
52*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[7*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[
7*ArcSin[c*x]])/(1536*c^3*x^3*(1 - c^2*x^2)^2)))/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. 1290 vs. \(2 (583 ) = 1166\).
time = 0.51, size = 1291, normalized size = 2.26

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1291\)
default	\(\text {Expression too large to display}\)	\(1291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(-35/8*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*c^3*x^3-1/3*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3*arcsi
n(c*x)^2+35/16*a^2/d^3*ln(c*x+1)-1/3*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)-2/3*a*b/d^3/(c^4
*x^4-2*c^2*x^2+1)/c^3/x^3*arcsin(c*x)-1/3*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/c^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
-7/3*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/c/x*arcsin(c*x)^2+175/24*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*c*x+29
/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)*c^2*x^2-35/4*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*ar
csin(c*x)*c^3*x^3+29/12*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)+175/12*a*b/d^3/(c^4*x^4-2*c^2
*x^2+1)*arcsin(c*x)*c*x-19/3*b^2/d^3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3*a^2/d^3/c/x+19/3*a*b/d^3*ln(
I*c*x+(-c^2*x^2+1)^(1/2)-1)-19/3*a*b/d^3*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-14/3*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)/c/x
*arcsin(c*x)-9/4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-9/4*a*b/d^3/(c^4*x^4-2*c^2*x^2+1
)*(-c^2*x^2+1)^(1/2)-35/4*a*b/d^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*a*b/d^3*arcsin(c*x)*ln(1
-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5/12*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*c^3*x^3+3/4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*c*
x-35/16*a^2/d^3*ln(c*x-1)-1/16*a^2/d^3/(c*x+1)^2-11/16*a^2/d^3/(c*x+1)+1/16*a^2/d^3/(c*x-1)^2-11/16*a^2/d^3/(c
*x-1)+35/4*b^2/d^3*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/4*b^2/d^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)
))-17/3*I*b^2/d^3*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-1/3*a^2/d^3/c^3/x^3+19/3*I*b^2/d^3*dilog(I*c*x+(-c^2*x^2+1)
^(1/2))+19/3*I*b^2/d^3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-35/4*I*b^2/d^3*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x
^2+1)^(1/2)))+35/4*I*b^2/d^3*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-1/3*b^2/d^3/(c^4*x^4-2*c^2*x
^2+1)/c/x-35/4*I*a*b/d^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*I*a*b/d^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1
/2)))+35/8*b^2/d^3*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/8*b^2/d^3*arcsin(c*x)^2*ln(1+I*(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((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/48*a^2*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 - 2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)
/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)) + 1/48*(105*(b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 105*(b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*arctan2(c*x, s
qrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(105*b^2*c^6*x^6 - 175*b^2*c^4*x^4 + 56*b^2*c^2*x^2 + 8*b^2)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 48*(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)*integrate(-1/24*(48*
a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (105*(b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 105*(b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*arctan2(c*x, sqr
t(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(105*b^2*c^7*x^7 - 175*b^2*c^5*x^5 + 56*b^2*c^3*x^3 + 8*b^2*c*x)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^
2*d^3*x^6 - d^3*x^4), x))/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^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((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^
4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {a^{2}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a**2/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integral(b**2*asin(c*x)**2/(c**6*x**10 -
3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integral(2*a*b*asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x*
*4), x))/d**3

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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