3.2.71 \(\int \frac {(d-c^2 d x^2)^2 (a+b \text {ArcSin}(c x))^2}{x^2} \, dx\) [171]

Optimal. Leaf size=249 \[ \frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {8}{3} c^2 d^2 x (a+b \text {ArcSin}(c x))^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{x}-4 b c d^2 (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )+2 i b^2 c d^2 \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d^2 \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right ) \]

[Out]

32/9*b^2*c^2*d^2*x-2/27*b^2*c^4*d^2*x^3-2/9*b*c*d^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-8/3*c^2*d^2*x*(a+b*ar
csin(c*x))^2-4/3*c^2*d^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2-d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x-4*b*c*d^2
*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*d^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^
2*c*d^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-10/3*b*c*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.34, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4785, 4743, 4715, 4767, 8, 4787, 4783, 4803, 4268, 2317, 2438} \begin {gather*} -\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{x}-\frac {8}{3} c^2 d^2 x (a+b \text {ArcSin}(c x))^2-4 b c d^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+2 i b^2 c d^2 \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d^2 \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )-\frac {2}{27} b^2 c^4 d^2 x^3+\frac {32}{9} b^2 c^2 d^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*b^2*c^2*d^2*x)/9 - (2*b^2*c^4*d^2*x^3)/27 - (10*b*c*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/3 - (2*b*c*
d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (8*c^2*d^2*x*(a + b*ArcSin[c*x])^2)/3 - (4*c^2*d^2*x*(1 - c^2
*x^2)*(a + b*ArcSin[c*x])^2)/3 - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - 4*b*c*d^2*(a + b*ArcSin[c*x])
*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^2*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*b^2*c*d^2*PolyLog[2, E^(I
*ArcSin[c*x])]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (4 c^2 d\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c d^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=\frac {2}{3} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{3} \left (8 c^2 d^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} \left (2 b^2 c^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac {1}{3} \left (8 b c^3 d^2\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {2}{3} b^2 c^2 d^2 x+\frac {2}{9} b^2 c^4 d^2 x^3+2 b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx+\frac {1}{9} \left (8 b^2 c^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^2\right ) \int 1 \, dx+\frac {1}{3} \left (16 b c^3 d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {16}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^2\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{3} \left (16 b^2 c^2 d^2\right ) \int 1 \, dx\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d^2\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (2 b^2 c d^2\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (2 i b^2 c d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=\frac {32}{9} b^2 c^2 d^2 x-\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {8}{3} c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4}{3} c^2 d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d^2 \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^2 \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.64, size = 322, normalized size = 1.29 \begin {gather*} \frac {1}{54} d^2 \left (-\frac {54 a^2}{x}-108 a^2 c^2 x+18 a^2 c^4 x^3+12 a b c \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+36 a b c^4 x^3 \text {ArcSin}(c x)-189 b^2 c \sqrt {1-c^2 x^2} \text {ArcSin}(c x)-216 a b c \left (\sqrt {1-c^2 x^2}+c x \text {ArcSin}(c x)\right )-108 b^2 c^2 x \left (-2+\text {ArcSin}(c x)^2\right )+2 b^2 c^2 x \left (-2 \left (6+c^2 x^2\right )+9 c^2 x^2 \text {ArcSin}(c x)^2\right )-\frac {108 a b \left (\text {ArcSin}(c x)+c x \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\right )}{x}-3 b^2 c \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x))-\frac {54 b^2 \text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 c x \left (-\log \left (1-e^{i \text {ArcSin}(c x)}\right )+\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )\right )}{x}+108 i b^2 c \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-108 i b^2 c \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*((-54*a^2)/x - 108*a^2*c^2*x + 18*a^2*c^4*x^3 + 12*a*b*c*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 36*a*b*c^4*x^3
*ArcSin[c*x] - 189*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 216*a*b*c*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) - 108
*b^2*c^2*x*(-2 + ArcSin[c*x]^2) + 2*b^2*c^2*x*(-2*(6 + c^2*x^2) + 9*c^2*x^2*ArcSin[c*x]^2) - (108*a*b*(ArcSin[
c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]))/x - 3*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (54*b^2*ArcSin[c*x]*(ArcS
in[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])))/x + (108*I)*b^2*c*PolyLog[2, -E^(
I*ArcSin[c*x])] - (108*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])]))/54

________________________________________________________________________________________

Maple [A]
time = 0.33, size = 372, normalized size = 1.49

method	result	size
derivativedivides	\(c \left (d^{2} a^{2} \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )-\frac {7 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {7 d^{2} b^{2} \arcsin \left (c x \right )^{2} c x}{4}+\frac {7 d^{2} b^{2} c x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{2} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{12}+\frac {d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{54}+2 d^{2} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(372\)
default	\(c \left (d^{2} a^{2} \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )-\frac {7 d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {7 d^{2} b^{2} \arcsin \left (c x \right )^{2} c x}{4}+\frac {7 d^{2} b^{2} c x}{2}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d^{2} b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{2} b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d^{2} b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{2} b^{2} \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{18}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{12}+\frac {d^{2} b^{2} \sin \left (3 \arcsin \left (c x \right )\right )}{54}+2 d^{2} a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(d^2*a^2*(1/3*c^3*x^3-2*c*x-1/c/x)-7/2*d^2*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-7/4*d^2*b^2*arcsin(c*x)^2*c*x+
7/2*d^2*b^2*c*x-d^2*b^2/c/x*arcsin(c*x)^2+2*d^2*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*d^2*b^2*arcsi
n(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*d^2*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*I*d^2*b^2*polylog(2,-I
*c*x-(-c^2*x^2+1)^(1/2))-1/18*d^2*b^2*arcsin(c*x)*cos(3*arcsin(c*x))-1/12*d^2*b^2*arcsin(c*x)^2*sin(3*arcsin(c
*x))+1/54*d^2*b^2*sin(3*arcsin(c*x))+2*d^2*a*b*(1/3*c^3*x^3*arcsin(c*x)-2*c*x*arcsin(c*x)-1/c/x*arcsin(c*x)+1/
9*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/9*(-c^2*x^2+1)^(1/2)-arctanh(1/(-c^2*x^2+1)^(1/2))))

________________________________________________________________________________________

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

[Out]

1/3*a^2*c^4*d^2*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^4*d^2 - 2*b^2*c^2*d^2*x*arcsin(c*x)^2 + 4*b^2*c^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) - 2*a^2*c^2*d^2
*x - 4*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d^2 - 2*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + a
rcsin(c*x)/x)*a*b*d^2 - a^2*d^2/x + 1/3*((b^2*c^4*d^2*x^4 - 3*b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x +
1))^2 + 3*x*integrate(2/3*(b^2*c^5*d^2*x^4 - 3*b^2*c*d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1))/(c^2*x^3 - x), x))/x

________________________________________________________________________________________

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)^2*(a+b*arcsin(c*x))^2/x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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 )}^2}{x^2} \,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)^2)/x^2,x)

[Out]

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

________________________________________________________________________________________