\(\int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx\) [213]

Optimal result

Integrand size = 23, antiderivative size = 439 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^4} \]

[Out]

-2*b^2*(a+b*arcsin(d*x+c))^2/d/e^4/(d*x+c)-1/3*(a+b*arcsin(d*x+c))^4/d/e^4/(d*x+c)^3-8*b^3*(a+b*arcsin(d*x+c))
*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-4/3*b*(a+b*arcsin(d*x+c))^3*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2
))/d/e^4+4*I*b^4*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4+2*I*b^2*(a+b*arcsin(d*x+c))^2*polylog(2,-I*(d
*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4-4*I*b^4*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-2*I*b^2*(a+b*arcsin(d*
x+c))^2*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-4*b^3*(a+b*arcsin(d*x+c))*polylog(3,-I*(d*x+c)-(1-(d*x+
c)^2)^(1/2))/d/e^4+4*b^3*(a+b*arcsin(d*x+c))*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-4*I*b^4*polylog(4,
-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4+4*I*b^4*polylog(4,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-2/3*b*(a+b*arcsin
(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d/e^4/(d*x+c)^2

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4723, 4789, 4803, 4268, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {8 b^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e^4}-\frac {4 b \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3}{3 d e^4}-\frac {4 b^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e^4}+\frac {4 b^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e^4}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{d e^4}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{d e^4}-\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^4} \]

[In]

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

[Out]

(-2*b^2*(a + b*ArcSin[c + d*x])^2)/(d*e^4*(c + d*x)) - (2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(
3*d*e^4*(c + d*x)^2) - (a + b*ArcSin[c + d*x])^4/(3*d*e^4*(c + d*x)^3) - (8*b^3*(a + b*ArcSin[c + d*x])*ArcTan
h[E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*b*(a + b*ArcSin[c + d*x])^3*ArcTanh[E^(I*ArcSin[c + d*x])])/(3*d*e^4) +
((4*I)*b^4*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^4) + ((2*I)*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[2, -E^(
I*ArcSin[c + d*x])])/(d*e^4) - ((4*I)*b^4*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^4) - ((2*I)*b^2*(a + b*ArcSi
n[c + d*x])^2*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, -E^(I*Arc
Sin[c + d*x])])/(d*e^4) + (4*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^4) - ((4*I)*b
^4*PolyLog[4, -E^(I*ArcSin[c + d*x])])/(d*e^4) + ((4*I)*b^4*PolyLog[4, E^(I*ArcSin[c + d*x])])/(d*e^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 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 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 4723

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

Rule 4789

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^4}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^4}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {(4 b) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{x^3 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e^4} \\ & = -\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {(2 b) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{x \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^2}{x^2} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}+\frac {(2 b) \text {Subst}\left (\int (a+b x)^3 \csc (x) \, dx,x,\arcsin (c+d x)\right )}{3 d e^4}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {a+b \arcsin (x)}{x \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}+\frac {\left (4 b^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c+d x))}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {\left (4 i b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}+\frac {\left (4 i b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {\left (4 i b^4\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {\left (4 i b^4\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4}-\frac {\left (4 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {\left (4 i b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {\left (4 i b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arcsin (c+d x)}\right )}{d e^4} \\ & = -\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^4} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1274\) vs. \(2(439)=878\).

Time = 11.49 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {a^4}{3 d e^4 (c+d x)^3}+\frac {a^2 b^2 \left (8 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-\frac {2 \left (2+4 \arcsin (c+d x)^2-2 \cos (2 \arcsin (c+d x))-3 (c+d x) \arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right )+3 (c+d x) \arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right )+4 i (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+2 \arcsin (c+d x) \sin (2 \arcsin (c+d x))+\arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right ) \sin (3 \arcsin (c+d x))-\arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right ) \sin (3 \arcsin (c+d x))\right )}{(c+d x)^3}\right )}{4 d e^4}+\frac {a b^3 \left (-24 \arcsin (c+d x) \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-4 \arcsin (c+d x)^3 \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-6 \arcsin (c+d x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-(c+d x) \arcsin (c+d x)^3 \csc ^4\left (\frac {1}{2} \arcsin (c+d x)\right )+24 \arcsin (c+d x)^2 \log \left (1-e^{i \arcsin (c+d x)}\right )-24 \arcsin (c+d x)^2 \log \left (1+e^{i \arcsin (c+d x)}\right )+48 \log \left (\tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )+48 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-48 i \arcsin (c+d x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )+48 \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )+6 \arcsin (c+d x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {16 \arcsin (c+d x)^3 \sin ^4\left (\frac {1}{2} \arcsin (c+d x)\right )}{(c+d x)^3}-24 \arcsin (c+d x) \tan \left (\frac {1}{2} \arcsin (c+d x)\right )-4 \arcsin (c+d x)^3 \tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )}{12 d e^4}+\frac {b^4 \left (-2 i \pi ^4+4 i \arcsin (c+d x)^4-24 \arcsin (c+d x)^2 \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-2 \arcsin (c+d x)^4 \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-4 \arcsin (c+d x)^3 \csc ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{2} (c+d x) \arcsin (c+d x)^4 \csc ^4\left (\frac {1}{2} \arcsin (c+d x)\right )+16 \arcsin (c+d x)^3 \log \left (1-e^{-i \arcsin (c+d x)}\right )+96 \arcsin (c+d x) \log \left (1-e^{i \arcsin (c+d x)}\right )-96 \arcsin (c+d x) \log \left (1+e^{i \arcsin (c+d x)}\right )-16 \arcsin (c+d x)^3 \log \left (1+e^{i \arcsin (c+d x)}\right )+48 i \arcsin (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (c+d x)}\right )+48 i \left (2+\arcsin (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-96 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+96 \arcsin (c+d x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (c+d x)}\right )-96 \arcsin (c+d x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )-96 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (c+d x)}\right )-96 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )+4 \arcsin (c+d x)^3 \sec ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {8 \arcsin (c+d x)^4 \sin ^4\left (\frac {1}{2} \arcsin (c+d x)\right )}{(c+d x)^3}-24 \arcsin (c+d x)^2 \tan \left (\frac {1}{2} \arcsin (c+d x)\right )-2 \arcsin (c+d x)^4 \tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )}{24 d e^4}+\frac {4 a^3 b \left (-\frac {1}{12} \arcsin (c+d x) \cot \left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{24} \csc ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{24} \arcsin (c+d x) \cot \left (\frac {1}{2} \arcsin (c+d x)\right ) \csc ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{6} \log \left (\cos \left (\frac {1}{2} \arcsin (c+d x)\right )\right )+\frac {1}{6} \log \left (\sin \left (\frac {1}{2} \arcsin (c+d x)\right )\right )+\frac {1}{24} \sec ^2\left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{12} \arcsin (c+d x) \tan \left (\frac {1}{2} \arcsin (c+d x)\right )-\frac {1}{24} \arcsin (c+d x) \sec ^2\left (\frac {1}{2} \arcsin (c+d x)\right ) \tan \left (\frac {1}{2} \arcsin (c+d x)\right )\right )}{d e^4} \]

[In]

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

[Out]

-1/3*a^4/(d*e^4*(c + d*x)^3) + (a^2*b^2*((8*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*(2 + 4*ArcSin[c + d*x]^
2 - 2*Cos[2*ArcSin[c + d*x]] - 3*(c + d*x)*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] + 3*(c + d*x)*ArcSin
[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + (4*I)*(c + d*x)^3*PolyLog[2, E^(I*ArcSin[c + d*x])] + 2*ArcSin[c +
d*x]*Sin[2*ArcSin[c + d*x]] + ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]] - ArcSin[c
+ d*x]*Log[1 + E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]]))/(c + d*x)^3))/(4*d*e^4) + (a*b^3*(-24*ArcSin[c
+ d*x]*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Cot[ArcSin[c + d*x]/2] - 6*ArcSin[c + d*x]^2*Csc[ArcSin[c
+ d*x]/2]^2 - (c + d*x)*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^4 + 24*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin
[c + d*x])] - 24*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] + 48*Log[Tan[ArcSin[c + d*x]/2]] + (48*I)*Ar
cSin[c + d*x]*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (48*I)*ArcSin[c + d*x]*PolyLog[2, E^(I*ArcSin[c + d*x])] -
48*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 48*PolyLog[3, E^(I*ArcSin[c + d*x])] + 6*ArcSin[c + d*x]^2*Sec[ArcSin[
c + d*x]/2]^2 - (16*ArcSin[c + d*x]^3*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]*Tan[ArcSin[c
+ d*x]/2] - 4*ArcSin[c + d*x]^3*Tan[ArcSin[c + d*x]/2]))/(12*d*e^4) + (b^4*((-2*I)*Pi^4 + (4*I)*ArcSin[c + d*x
]^4 - 24*ArcSin[c + d*x]^2*Cot[ArcSin[c + d*x]/2] - 2*ArcSin[c + d*x]^4*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c +
d*x]^3*Csc[ArcSin[c + d*x]/2]^2 - ((c + d*x)*ArcSin[c + d*x]^4*Csc[ArcSin[c + d*x]/2]^4)/2 + 16*ArcSin[c + d*x
]^3*Log[1 - E^((-I)*ArcSin[c + d*x])] + 96*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] - 96*ArcSin[c + d*x]
*Log[1 + E^(I*ArcSin[c + d*x])] - 16*ArcSin[c + d*x]^3*Log[1 + E^(I*ArcSin[c + d*x])] + (48*I)*ArcSin[c + d*x]
^2*PolyLog[2, E^((-I)*ArcSin[c + d*x])] + (48*I)*(2 + ArcSin[c + d*x]^2)*PolyLog[2, -E^(I*ArcSin[c + d*x])] -
(96*I)*PolyLog[2, E^(I*ArcSin[c + d*x])] + 96*ArcSin[c + d*x]*PolyLog[3, E^((-I)*ArcSin[c + d*x])] - 96*ArcSin
[c + d*x]*PolyLog[3, -E^(I*ArcSin[c + d*x])] - (96*I)*PolyLog[4, E^((-I)*ArcSin[c + d*x])] - (96*I)*PolyLog[4,
-E^(I*ArcSin[c + d*x])] + 4*ArcSin[c + d*x]^3*Sec[ArcSin[c + d*x]/2]^2 - (8*ArcSin[c + d*x]^4*Sin[ArcSin[c +
d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]^2*Tan[ArcSin[c + d*x]/2] - 2*ArcSin[c + d*x]^4*Tan[ArcSin[c + d*x]
/2]))/(24*d*e^4) + (4*a^3*b*(-1/12*(ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2]) - Csc[ArcSin[c + d*x]/2]^2/24 - (A
rcSin[c + d*x]*Cot[ArcSin[c + d*x]/2]*Csc[ArcSin[c + d*x]/2]^2)/24 - Log[Cos[ArcSin[c + d*x]/2]]/6 + Log[Sin[A
rcSin[c + d*x]/2]]/6 + Sec[ArcSin[c + d*x]/2]^2/24 - (ArcSin[c + d*x]*Tan[ArcSin[c + d*x]/2])/12 - (ArcSin[c +
d*x]*Sec[ArcSin[c + d*x]/2]^2*Tan[ArcSin[c + d*x]/2])/24))/(d*e^4)

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 1009, normalized size of antiderivative = 2.30

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1009\)
default	\(\text {Expression too large to display}\)	\(1009\)
parts	\(\text {Expression too large to display}\)	\(1020\)

[In]

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

[Out]

1/d*(-1/3*a^4/e^4/(d*x+c)^3+b^4/e^4*(-1/3/(d*x+c)^3*arcsin(d*x+c)^2*(2*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+
c)+arcsin(d*x+c)^2+6*(d*x+c)^2)-2/3*arcsin(d*x+c)^3*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+2*I*arcsin(d*x+c)^2*po
lylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-4*arcsin(d*x+c)*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-4*I*polylog(
4,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+2/3*arcsin(d*x+c)^3*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d*x+c)^2*
polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+4*arcsin(d*x+c)*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+4*I*polylog(
4,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-4*arcsin(d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+4*I*polylog(2,-I*(d*x+c)-
(1-(d*x+c)^2)^(1/2))+4*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-4*I*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)
^(1/2)))+4*a*b^3/e^4*(-1/6/(d*x+c)^3*arcsin(d*x+c)*(3*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+2*arcsin(d*x+c
)^2+6*(d*x+c)^2)+1/2*arcsin(d*x+c)^2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-I*arcsin(d*x+c)*polylog(2,I*(d*x+c)+(
1-(d*x+c)^2)^(1/2))+polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-1/2*arcsin(d*x+c)^2*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^
(1/2))+I*arcsin(d*x+c)*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*a
rctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2)))+6*a^2*b^2/e^4*(-1/3*(arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+arcsin(d
*x+c)^2+(d*x+c)^2)/(d*x+c)^3+1/3*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-1/3*I*polylog(2,I*(d*x+c)+(
1-(d*x+c)^2)^(1/2))-1/3*arcsin(d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+1/3*I*polylog(2,-I*(d*x+c)-(1-(d*x+c
)^2)^(1/2)))+4*a^3*b/e^4*(-1/3/(d*x+c)^3*arcsin(d*x+c)-1/6/(d*x+c)^2*(1-(d*x+c)^2)^(1/2)-1/6*arctanh(1/(1-(d*x
+c)^2)^(1/2))))

Fricas [F]

\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*asin(d*x+c))**4/(d*e*x+c*e)**4,x)

[Out]

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

Maxima [F]

\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="maxima")

[Out]

-1/3*a^4/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*(b^4*arctan2(d*x + c, sqrt(d*x +
c + 1)*sqrt(-d*x - c + 1))^4 + 3*(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)*integrate(2/3*(
2*(b^4*d*x + b^4*c)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)
)^3 - 6*(a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^3*c^2 - a*b^3)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c +
1))^3 - 9*(a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + a^2*b^2*c^2 - a^2*b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt
(-d*x - c + 1))^2 - 6*(a^3*b*d^2*x^2 + 2*a^3*b*c*d*x + a^3*b*c^2 - a^3*b)*arctan2(d*x + c, sqrt(d*x + c + 1)*s
qrt(-d*x - c + 1)))/(d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + (15*c^2 - 1)*d^4*e^4*x^4 + 4*(5*c^3 - c)*d^3*e^4*x^3 + 3*
(5*c^4 - 2*c^2)*d^2*e^4*x^2 + 2*(3*c^5 - 2*c^3)*d*e^4*x + (c^6 - c^4)*e^4), x))/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2
+ 3*c^2*d^2*e^4*x + c^3*d*e^4)

Giac [F]

\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^4,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^4/(d*e*x + c*e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]

[In]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^4,x)

[Out]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^4, x)