3.2.82 \(\int \frac {(d-c^2 d x^2)^3 (a+b \text {ArcSin}(c x))^2}{x^4} \, dx\) [182]

Optimal. Leaf size=348 \[ -\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {ArcSin}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{3 x^3}+\frac {34}{3} b c^3 d^3 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )-\frac {17}{3} i b^2 c^3 d^3 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+\frac {17}{3} i b^2 c^3 d^3 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right ) \]

[Out]

-1/3*b^2*c^2*d^3/x-50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3-1/9*b*c^3*d^3*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-
1/3*b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/x^2+16/3*c^4*d^3*x*(a+b*arcsin(c*x))^2+8/3*c^4*d^3*x*(-c^2*x^
2+1)*(a+b*arcsin(c*x))^2+2*c^2*d^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x-1/3*d^3*(-c^2*x^2+1)^3*(a+b*arcsin(c*x
))^2/x^3+34/3*b*c^3*d^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))-17/3*I*b^2*c^3*d^3*polylog(2,-I*c*
x-(-c^2*x^2+1)^(1/2))+17/3*I*b^2*c^3*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+5*b*c^3*d^3*(a+b*arcsin(c*x))*(-c
^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.66, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4785, 4743, 4715, 4767, 8, 4787, 4783, 4803, 4268, 2317, 2438, 276} \begin {gather*} \frac {16}{3} c^4 d^3 x (a+b \text {ArcSin}(c x))^2+\frac {34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{x}-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{3 x^2}-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{3 x^3}+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+5 b c^3 d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {17}{3} i b^2 c^3 d^3 \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )+\frac {17}{3} i b^2 c^3 d^3 \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )+\frac {2}{27} b^2 c^6 d^3 x^3-\frac {50}{9} b^2 c^4 d^3 x-\frac {b^2 c^2 d^3}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b^2*c^2*d^3)/x - (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 + 5*b*c^3*d^3*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x]) - (b*c^3*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSi
n[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSin[c*x])^2)/3 + (8*c^4*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/
3 + (2*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(3*x^3)
+ (34*b*c^3*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, -E^(I*Ar
cSin[c*x])] + ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, E^(I*ArcSin[c*x])]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 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 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 4715

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

Rule 4743

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((
a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*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*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 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 4783

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f
*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(f*x)^(m + 1)*(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] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4785

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*((a + b*ArcSin[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(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]) /; FreeQ[{a, b, c,
d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 4787

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*((a + b*ArcSin[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\left (2 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \frac {\left (1-c^2 x^2\right )^2}{x^2} \, dx-\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\left (4 b c^3 d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=-\frac {17}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \left (-2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx-\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\left (4 b c^3 d^3\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac {1}{3} \left (16 c^4 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac {1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac {1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac {1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {11}{9} b^2 c^4 d^3 x-\frac {14}{27} b^2 c^6 d^3 x^3-\frac {17}{3} b c^3 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{3} \left (5 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx-\left (4 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx+\frac {1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx-\frac {1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac {1}{3} \left (32 b c^5 d^3\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {46}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{3} \left (5 b c^3 d^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\left (4 b c^3 d^3\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{3} \left (5 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac {1}{3} \left (5 i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac {1}{3} \left (5 i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (4 i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\left (4 i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac {17}{3} i b^2 c^3 d^3 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac {17}{3} i b^2 c^3 d^3 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 480, normalized size = 1.38 \begin {gather*} -\frac {d^3 \left (9 a^2-81 a^2 c^2 x^2+9 b^2 c^2 x^2-81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6-2 b^2 c^6 x^6+9 a b c x \sqrt {1-c^2 x^2}-150 a b c^3 x^3 \sqrt {1-c^2 x^2}+6 a b c^5 x^5 \sqrt {1-c^2 x^2}+18 a b \text {ArcSin}(c x)-162 a b c^2 x^2 \text {ArcSin}(c x)-162 a b c^4 x^4 \text {ArcSin}(c x)+18 a b c^6 x^6 \text {ArcSin}(c x)+9 b^2 c x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)-150 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+6 b^2 c^5 x^5 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+9 b^2 \text {ArcSin}(c x)^2-81 b^2 c^2 x^2 \text {ArcSin}(c x)^2-81 b^2 c^4 x^4 \text {ArcSin}(c x)^2+9 b^2 c^6 x^6 \text {ArcSin}(c x)^2-153 a b c^3 x^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )+153 b^2 c^3 x^3 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-153 b^2 c^3 x^3 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+153 i b^2 c^3 x^3 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-153 i b^2 c^3 x^3 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*(d^3*(9*a^2 - 81*a^2*c^2*x^2 + 9*b^2*c^2*x^2 - 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 - 2*b^2*
c^6*x^6 + 9*a*b*c*x*Sqrt[1 - c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 6*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] +
18*a*b*ArcSin[c*x] - 162*a*b*c^2*x^2*ArcSin[c*x] - 162*a*b*c^4*x^4*ArcSin[c*x] + 18*a*b*c^6*x^6*ArcSin[c*x] +
9*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 150*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^5*x^5*Sqrt[1
- c^2*x^2]*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2 - 81*b^2*c^2*x^2*ArcSin[c*x]^2 - 81*b^2*c^4*x^4*ArcSin[c*x]^2 +
9*b^2*c^6*x^6*ArcSin[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] + 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 -
E^(I*ArcSin[c*x])] - 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (153*I)*b^2*c^3*x^3*PolyLog[2,
-E^(I*ArcSin[c*x])] - (153*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/x^3

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Maple [A]
time = 0.65, size = 488, normalized size = 1.40

method	result	size
derivativedivides	\(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}-\frac {50 d^{3} b^{2} c x}{9}-\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i d^{3} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 i d^{3} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x -\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}+\frac {50 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(488\)
default	\(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}-\frac {d^{3} b^{2}}{3 c x}-\frac {50 d^{3} b^{2} c x}{9}-\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i d^{3} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 d^{3} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {17 i d^{3} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {3 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arcsin \left (c x \right )^{2} c x -\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}+\frac {50 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {17 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(-d^3*a^2*(1/3*c^3*x^3-3*c*x+1/3/c^3/x^3-3/c/x)+2/27*d^3*b^2*c^3*x^3-1/3*d^3*b^2/c/x-50/9*d^3*b^2*c*x-17/3
*d^3*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+17/3*I*d^3*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+17/3*d^
3*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-17/3*I*d^3*b^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+3*d^3*b^2
/c/x*arcsin(c*x)^2-1/3*d^3*b^2/c^3/x^3*arcsin(c*x)^2-1/3*d^3*b^2*arcsin(c*x)^2*c^3*x^3+3*d^3*b^2*arcsin(c*x)^2
*c*x-1/3*d^3*b^2/c^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+50/9*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-2/9*d^3*b^
2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*d^3*a*b*(1/3*c^3*x^3*arcsin(c*x)-3*c*x*arcsin(c*x)+1/3/c^3/x^3*arcs
in(c*x)-3/c/x*arcsin(c*x)+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)-25/9*(-c^2*x^2+1)^(1/2)+1/6/c^2/x^2*(-c^2*x^2+1)^(1/2
)-17/6*arctanh(1/(-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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxima")

[Out]

-1/3*a^2*c^6*d^3*x^3 - 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b
*c^6*d^3 + 3*b^2*c^4*d^3*x*arcsin(c*x)^2 - 6*b^2*c^4*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + 3*a^2*c^4*d^
3*x + 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c^3*d^3 + 6*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x))
+ arcsin(c*x)/x)*a*b*c^2*d^3 - 1/3*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)
*c + 2*arcsin(c*x)/x^3)*a*b*d^3 + 3*a^2*c^2*d^3/x - 1/3*a^2*d^3/x^3 - 1/3*(3*x^3*integrate(2/3*(b^2*c^7*d^3*x^
6 - 9*b^2*c^3*d^3*x^2 + b^2*c*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^
2*x^5 - x^3), x) + (b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 + b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^
2)/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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^4,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\,{\left (d-c^2\,d\,x^2\right )}^3}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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