\(\int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx\) [202]

Optimal result

Integrand size = 23, antiderivative size = 169 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12, 4721, 3798, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{2 d e}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2}{2 d e}-\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {\log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3}{d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \]

[In]

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

[Out]

((-1/4*I)*(a + b*ArcSin[c + d*x])^4)/(b*d*e) + ((a + b*ArcSin[c + d*x])^3*Log[1 - E^((2*I)*ArcSin[c + d*x])])/
(d*e) - (((3*I)/2)*b*(a + b*ArcSin[c + d*x])^2*PolyLog[2, E^((2*I)*ArcSin[c + d*x])])/(d*e) + (3*b^2*(a + b*Ar
cSin[c + d*x])*PolyLog[3, E^((2*I)*ArcSin[c + d*x])])/(2*d*e) + (((3*I)/4)*b^3*PolyLog[4, E^((2*I)*ArcSin[c +
d*x])])/(d*e)

Rule 12

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+96 a^2 b \arcsin (c+d x)^2-64 a b^2 \arcsin (c+d x)^3-16 b^3 \arcsin (c+d x)^4+192 i a b^2 \arcsin (c+d x)^2 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+64 i b^3 \arcsin (c+d x)^3 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+192 i a^2 b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+64 i a^3 \log (c+d x)-96 b^2 \arcsin (c+d x) (2 a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c+d x)}\right )+96 a^2 b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )+96 i a b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+96 i b^3 \arcsin (c+d x) \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+48 b^3 \operatorname {PolyLog}\left (4,e^{-2 i \arcsin (c+d x)}\right )\right )}{64 d e} \]

[In]

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

[Out]

((-1/64*I)*(8*a*b^2*Pi^3 + b^3*Pi^4 + 96*a^2*b*ArcSin[c + d*x]^2 - 64*a*b^2*ArcSin[c + d*x]^3 - 16*b^3*ArcSin[
c + d*x]^4 + (192*I)*a*b^2*ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (64*I)*b^3*ArcSin[c + d*x]^
3*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (192*I)*a^2*b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + (64
*I)*a^3*Log[c + d*x] - 96*b^2*ArcSin[c + d*x]*(2*a + b*ArcSin[c + d*x])*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])]
+ 96*a^2*b*PolyLog[2, E^((2*I)*ArcSin[c + d*x])] + (96*I)*a*b^2*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] + (96*
I)*b^3*ArcSin[c + d*x]*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] + 48*b^3*PolyLog[4, E^((-2*I)*ArcSin[c + d*x])])
)/(d*e)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (206 ) = 412\).

Time = 0.71 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.86

method	result	size
derivativedivides	\(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(652\)
default	\(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(652\)
parts	\(\frac {a^{3} \ln \left (d x +c \right )}{e d}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\)	\(660\)

[In]

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

[Out]

1/d*(a^3/e*ln(d*x+c)+b^3/e*(-1/4*I*arcsin(d*x+c)^4+arcsin(d*x+c)^3*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-3*I*arc
sin(d*x+c)^2*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+6*arcsin(d*x+c)*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))
+6*I*polylog(4,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+arcsin(d*x+c)^3*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-3*I*arcsin(d
*x+c)^2*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+6*arcsin(d*x+c)*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+6*
I*polylog(4,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2)))+3*a*b^2/e*(-1/3*I*arcsin(d*x+c)^3+arcsin(d*x+c)^2*ln(1+I*(d*x+c)+
(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d*x+c)*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+2*polylog(3,-I*(d*x+c)-(1-(d*
x+c)^2)^(1/2))+arcsin(d*x+c)^2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d*x+c)*polylog(2,I*(d*x+c)+(1-(d
*x+c)^2)^(1/2))+2*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2)))+3*a^2*b/e*(-1/2*I*arcsin(d*x+c)^2+arcsin(d*x+c)*ln
(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-I*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-
(d*x+c)^2)^(1/2))-I*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))))

Fricas [F]

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

[In]

integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e),x, algorithm="maxima")

[Out]

a^3*log(d*e*x + c*e)/(d*e) + integrate((b^3*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*a*b^2
*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 3*a^2*b*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d
*x - c + 1)))/(d*e*x + c*e), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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