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

Optimal result

Integrand size = 27, antiderivative size = 296 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \]

[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)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4793, 4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 4747, 267} \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=-\frac {2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^3}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3} \]

[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*} \text {integral}& = \frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^2} \, dx}{d} \\ & = -\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {(b c) \int \frac {a+b \arcsin (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 {(a+b \arcsin (c x))^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 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^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,\arcsin (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 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^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,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (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,\arcsin (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,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \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 \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \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 \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^3} \\ & = \frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \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 \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(756\) vs. \(2(296)=592\).

Time = 4.56 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2}{\left (-1+c^2 x^2\right )^2}-\frac {12 a^2}{-1+c^2 x^2}+\frac {15 a b \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{-1+c x}+\frac {15 a b \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{1+c x}+\frac {a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(-1+c x)^2}+\frac {a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(1+c x)^2}+48 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+24 a^2 \log (c x)-12 a^2 \log \left (1-c^2 x^2\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-3 i \pi -4 \log \left (1+i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )+\log \left (1+i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-i \pi -4 \log \left (1-i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )-\log \left (1-i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-24 i a b \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-b^2 \left (i \pi ^3+\frac {2}{-1+c^2 x^2}+\frac {4 c x \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {32 c x \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 \arcsin (c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 \arcsin (c x)^2}{-1+c^2 x^2}-16 i \arcsin (c x)^3-24 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+24 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+16 \log \left (1-c^2 x^2\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )}{24 d^3} \]

[In]

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

[Out]

((6*a^2)/(-1 + c^2*x^2)^2 - (12*a^2)/(-1 + c^2*x^2) + (15*a*b*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(-1 + c*x) +
(15*a*b*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(1 + c*x) + (a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(-
1 + c*x)^2 + (a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(1 + c*x)^2 + 48*a*b*ArcSin[c*x]*Log[1 - E^((
2*I)*ArcSin[c*x])] + 24*a^2*Log[c*x] - 12*a^2*Log[1 - c^2*x^2] + 12*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-3*I)
*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) + 2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])]
+ 2*Log[Cos[ArcSin[c*x]/2]] - Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) +
12*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-I)*Pi - 4*Log[1 - I*E^(I*ArcSin[c*x])]) + 2*Pi*(-2*Log[1 + E^((-I)*A
rcSin[c*x])] - Log[1 - I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] + Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) +
(4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) - (24*I)*a*b*(ArcSin[c*x]^2 + 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])]))/(24*d^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (324 ) = 648\).

Time = 0.44 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.56

method	result	size
parts	\(-\frac {a^{2} \left (-\ln \left (x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\)	\(759\)
derivativedivides	\(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\)	\(761\)
default	\(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\)	\(761\)

[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(x)-1/16/(c*x-1)^2+5/16/(c*x-1)+1/2*ln(c*x-1)-1/16/(c*x+1)^2-5/16/(c*x+1)+1/2*ln(c*x+1))-b^2/d^3*
(1/12*(16*I*arcsin(c*x)*c^4*x^4-16*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3+6*arcsin(c*x)^2*x^2*c^2-32*I*arcsin(
c*x)*c^2*x^2+18*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-9*arcsin(c*x)^2+16*I*arcsin(c*x)+c^2*x^2-1)/(c^4*x^4-2*c^2*
x^2+1)+4/3*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-8/3*ln(I*c*x+(-c^2*x^2+1)^(1/2))-arcsin(c*x)^2*ln(1+I*c*x+(-c^2*
x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+arcs
in(c*x)^2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*polylo
g(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,I*c*
x+(-c^2*x^2+1)^(1/2))-2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(1/12*(8*I*c^4*x^4-8*c^3*x^3*(-c^2*x^2+
1)^(1/2)+6*c^2*x^2*arcsin(c*x)-16*I*c^2*x^2+9*c*x*(-c^2*x^2+1)^(1/2)-9*arcsin(c*x)+8*I)/(c^4*x^4-2*c^2*x^2+1)-
arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1+(I*c*x+(-c^
2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+I
*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \]

[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

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

[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)