3.209 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{x^4 (d-c^2 d x^2)^3} \, dx\)

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

[Out]

-(b^2*c^2)/(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*ArcTanh[
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)

________________________________________________________________________________________

Rubi [A]  time = 1.32127, antiderivative size = 572, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 17, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.63, Rules used = {4701, 4655, 4657, 4181, 2531, 2282, 6589, 4677, 206, 199, 4705, 4709, 4183, 2279, 2391, 290, 325} \[ \frac{35 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}-\frac{35 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}+\frac{19 i b^2 c^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac{19 i b^2 c^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{3 d^3}-\frac{35 b^2 c^3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}+\frac{35 b^2 c^3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 d^3}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{29 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{12 d^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{7 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{35 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3}-\frac{38 b c^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^3}-\frac{b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac{b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac{b^2 c^2}{2 d^3 x}+\frac{17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c^2)/(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*ArcTanh[
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 4701

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*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4655

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[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 4657

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 4181

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 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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 206

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

Rule 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 4705

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*d^IntPart[p]*(d + e*x^2)^Frac
Part[p])/(2*f*(p + 1)*(1 - c^2*x^2)^FracPart[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, 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 4709

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

Rule 4183

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 2279

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 2391

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 290

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] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [B]  time = 12.0335, size = 1657, normalized size = 2.9 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-a^2/(3*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)*Sq
rt[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*ArcSi
n[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*Arc
Sin[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*Ar
cSin[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)^(1/4)*(-I + E^(I*ArcSin[c*x])))/(2*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]*Lo
g[-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[Cos[
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*Log
[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)*ArcS
in[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 210*PolyLog
[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*ArcSin
[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*Ar
cSin[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*A
rcSin[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]] - 116*
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]] + 152
*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]  time = 0.494, size = 1352, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-35/4*I*c^3*a*b/d^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*I*c^3*b^2/d^3*arcsin(c*x)*polylog(2,-I*(I*c*x+(
-c^2*x^2+1)^(1/2)))-35/4*I*c^3*b^2/d^3*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*I*c^3*a*b/d^3*
dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-7/3*c^2*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/x*arcsin(c*x)^2-35/8*c^6*b^2/d^3/(
c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*x^3+175/24*c^4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*x-9/4*c^3*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*c^3*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(-c^2*x^2+1)^(1/2
)-35/4*c^3*a*b/d^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*c^3*a*b/d^3*arcsin(c*x)*ln(1-I*(I*c*x+(
-c^2*x^2+1)^(1/2)))-2/3*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x^3*arcsin(c*x)-1/3*c*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*
(-c^2*x^2+1)^(1/2)-1/3*c*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/3*a^2/d^3/x^3-35/1
6*c^3*a^2/d^3*ln(c*x-1)+35/16*c^3*a^2/d^3*ln(c*x+1)+1/16*c^3*a^2/d^3/(c*x-1)^2-11/16*c^3*a^2/d^3/(c*x-1)-1/16*
c^3*a^2/d^3/(c*x+1)^2-11/16*c^3*a^2/d^3/(c*x+1)-3*c^2*a^2/d^3/x-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*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3+19/3*I*c^3*b^2/d^3*dilog(1+I*c*x+(-c^2*x^
2+1)^(1/2))-17/3*I*c^3*b^2/d^3*arctan(I*c*x+(-c^2*x^2+1)^(1/2))+19/3*I*c^3*b^2/d^3*dilog(I*c*x+(-c^2*x^2+1)^(1
/2))-1/3*c^2*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/x+3/4*c^4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*x-5/12*c^6*b^2/d^3/(c^4*x^4
-2*c^2*x^2+1)*x^3-1/3*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)/x^3*arcsin(c*x)^2-19/3*c^3*b^2/d^3*arcsin(c*x)*ln(1+I*c*x+
(-c^2*x^2+1)^(1/2))-35/8*c^3*b^2/d^3*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/8*c^3*b^2/d^3*arcsin(
c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+19/3*c^3*a*b/d^3*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-19/3*c^3*a*b/d^3*ln(
1+I*c*x+(-c^2*x^2+1)^(1/2))-14/3*c^2*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x*arcsin(c*x)+29/12*c^5*b^2/d^3/(c^4*x^4-2*
c^2*x^2+1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2-35/4*c^6*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*x^3+29/12*c^5
*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*(-c^2*x^2+1)^(1/2)+175/12*c^4*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{48} \, a^{2}{\left (\frac{105 \, c^{3} \log \left (c x + 1\right )}{d^{3}} - \frac{105 \, c^{3} \log \left (c x - 1\right )}{d^{3}} - \frac{2 \,{\left (105 \, c^{6} x^{6} - 175 \, c^{4} x^{4} + 56 \, c^{2} x^{2} + 8\right )}}{c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}\right )} + \frac{105 \,{\left (b^{2} c^{7} x^{7} - 2 \, b^{2} c^{5} x^{5} + b^{2} c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) - 105 \,{\left (b^{2} c^{7} x^{7} - 2 \, b^{2} c^{5} x^{5} + b^{2} c^{3} x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \,{\left (105 \, b^{2} c^{6} x^{6} - 175 \, b^{2} c^{4} x^{4} + 56 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - 2 \,{\left (c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}\right )} \int \frac{48 \, a b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (105 \,{\left (b^{2} c^{8} x^{8} - 2 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 105 \,{\left (b^{2} c^{8} x^{8} - 2 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (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\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \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}}\,{d x}}{48 \,{\left (c^{4} d^{3} x^{7} - 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}\right )}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + 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\right ) \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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