3.3.6 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x (d-c^2 d x^2)^3} \, dx\) [206]

Optimal. Leaf size=296 \[ \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \text {ArcSin}(c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \text {ArcSin}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \text {ArcSin}(c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d^3}-\frac {i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{d^3}-\frac {b^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {b^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3} \]

[Out]

1/12*b^2/d^3/(-c^2*x^2+1)-1/6*b*c*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(3/2)+1/4*(a+b*arcsin(c*x))^2/d^3/(-c^2
*x^2+1)^2+1/2*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)-2*(a+b*arcsin(c*x))^2*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)
/d^3-2/3*b^2*ln(-c^2*x^2+1)/d^3+I*b*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-I*b*(a+b*ar
csin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-1/2*b^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+1/
2*b^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-4/3*b*c*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 296, 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 = {4793, 4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 4747, 267} \begin {gather*} -\frac {4 b c x (a+b \text {ArcSin}(c x))}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {b c x (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \text {ArcSin}(c x))^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac {(a+b \text {ArcSin}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {i b \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^3}-\frac {i b \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^3}-\frac {2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^3}-\frac {b^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {b^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b^2/(12*d^3*(1 - c^2*x^2)) - (b*c*x*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (4*b*c*x*(a + b*ArcSin[
c*x]))/(3*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])^2/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcSin[c*x])^2/(2*d^
3*(1 - c^2*x^2)) - (2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d^3 - (2*b^2*Log[1 - c^2*x^2])/(3*
d^3) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d^3 - (I*b*(a + b*ArcSin[c*x])*PolyLog[2,
E^((2*I)*ArcSin[c*x])])/d^3 - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d^3) + (b^2*PolyLog[3, E^((2*I)*ArcS
in[c*x])])/(2*d^3)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 267

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

Rule 2320

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

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 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 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4745

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSin[c
*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcSin
[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 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 4769

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

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

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Mathematica [A]
time = 2.36, size = 459, normalized size = 1.55 \begin {gather*} -\frac {-\frac {6 a^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 a^2}{-1+c^2 x^2}-24 a^2 \log (c x)+12 a^2 \log \left (1-c^2 x^2\right )+4 a b \left (\frac {c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac {8 c x}{\sqrt {1-c^2 x^2}}-\frac {3 \text {ArcSin}(c x)}{\left (-1+c^2 x^2\right )^2}+\frac {6 \text {ArcSin}(c x)}{-1+c^2 x^2}-12 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+12 \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-6 i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+6 i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+b^2 \left (i \pi ^3+\frac {2}{-1+c^2 x^2}+\frac {4 c x \text {ArcSin}(c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {32 c x \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-\frac {6 \text {ArcSin}(c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 \text {ArcSin}(c x)^2}{-1+c^2 x^2}-16 i \text {ArcSin}(c x)^3-24 \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+24 \text {ArcSin}(c x)^2 \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )+16 \log \left (1-c^2 x^2\right )-24 i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )-24 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-12 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )+12 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )\right )}{24 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*((-6*a^2)/(-1 + c^2*x^2)^2 + (12*a^2)/(-1 + c^2*x^2) - 24*a^2*Log[c*x] + 12*a^2*Log[1 - c^2*x^2] + 4*a*b
*((c*x)/(1 - c^2*x^2)^(3/2) + (8*c*x)/Sqrt[1 - c^2*x^2] - (3*ArcSin[c*x])/(-1 + c^2*x^2)^2 + (6*ArcSin[c*x])/(
-1 + c^2*x^2) - 12*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 12*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])]
- (6*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (6*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]) + b^2*(I*Pi^3 + 2/(-1 +
c^2*x^2) + (4*c*x*ArcSin[c*x])/(1 - c^2*x^2)^(3/2) + (32*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*ArcSin[c*x]^2
)/(-1 + c^2*x^2)^2 + (12*ArcSin[c*x]^2)/(-1 + c^2*x^2) - (16*I)*ArcSin[c*x]^3 - 24*ArcSin[c*x]^2*Log[1 - E^((-
2*I)*ArcSin[c*x])] + 24*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 16*Log[1 - c^2*x^2] - (24*I)*ArcSin[c*x
]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (24*I)*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 12*PolyLog[3, E
^((-2*I)*ArcSin[c*x])] + 12*PolyLog[3, -E^((2*I)*ArcSin[c*x])]))/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. 1223 vs. \(2 (324 ) = 648\).
time = 0.38, size = 1224, normalized size = 4.14

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1224\)
default	\(\text {Expression too large to display}\)	\(1224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-1/4*a^2*((2*c^2*x^2 - 3)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) + 2*log(c*x + 1)/d^3 + 2*log(c*x - 1)/d^3 - 4*lo
g(x)/d^3) - integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1)))/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)

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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/(-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^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x),
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^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a**2/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x) + Integral(b**2*asin(c*x)**2/(c**6*x**7 - 3*c*
*4*x**5 + 3*c**2*x**3 - x), x) + Integral(2*a*b*asin(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x))/d**
3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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\,{\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*(d - c^2*d*x^2)^3),x)

[Out]

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

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