3.2.9 \(\int \frac {(f+g x+h x^2+i x^3) (a+b \text {ArcSin}(c x))}{d+e x} \, dx\) [109]

Optimal. Leaf size=623 \[ \frac {b i x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (2 e^2 i+9 c^2 \left (e^2 g-d e h+d^2 i\right )\right )+9 c^2 e (e h-d i) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}-\frac {b (e h-d i) \text {ArcSin}(c x)}{4 c^2 e^2}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {ArcSin}(c x)^2}{2 e^4}+\frac {\left (e^2 g-d e h+d^2 i\right ) x (a+b \text {ArcSin}(c x))}{e^3}+\frac {(e h-d i) x^2 (a+b \text {ArcSin}(c x))}{2 e^2}+\frac {i x^3 (a+b \text {ArcSin}(c x))}{3 e}+\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {ArcSin}(c x) \log (d+e x)}{e^4}+\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]

[Out]

-1/4*b*(-d*i+e*h)*arcsin(c*x)/c^2/e^2-1/2*I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)^2/e^4+(d^2*i-d*e*h+e^
2*g)*x*(a+b*arcsin(c*x))/e^3+1/2*(-d*i+e*h)*x^2*(a+b*arcsin(c*x))/e^2+1/3*i*x^3*(a+b*arcsin(c*x))/e-b*(-d^3*i+
d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(e*x+d)/e^4+(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))*ln(e*x+d)/e^
4+b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/
e^4+b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2))
)/e^4-I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e
^4-I*b*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4+
1/9*b*i*x^2*(-c^2*x^2+1)^(1/2)/c/e+1/36*b*(8*e^2*i+36*c^2*(d^2*i-d*e*h+e^2*g)+9*c^2*e*(-d*i+e*h)*x)*(-c^2*x^2+
1)^(1/2)/c^3/e^3

________________________________________________________________________________________

Rubi [A]
time = 0.81, antiderivative size = 623, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {1864, 4837, 12, 6874, 1823, 794, 222, 2451, 4825, 4615, 2221, 2317, 2438} \begin {gather*} \frac {x (a+b \text {ArcSin}(c x)) \left (d^2 i-d e h+e^2 g\right )}{e^3}+\frac {\log (d+e x) (a+b \text {ArcSin}(c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {x^2 (e h-d i) (a+b \text {ArcSin}(c x))}{2 e^2}+\frac {i x^3 (a+b \text {ArcSin}(c x))}{3 e}-\frac {i b \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}-\frac {i b \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {b \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {b \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}-\frac {b \text {ArcSin}(c x) (e h-d i)}{4 c^2 e^2}-\frac {i b \text {ArcSin}(c x)^2 \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4}-\frac {b \text {ArcSin}(c x) \log (d+e x) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4}+\frac {b i x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (9 c^2 \left (d^2 i-d e h+e^2 g\right )+2 e^2 i\right )+9 c^2 e x (e h-d i)\right )}{36 c^3 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x),x]

[Out]

(b*i*x^2*Sqrt[1 - c^2*x^2])/(9*c*e) + (b*(4*(2*e^2*i + 9*c^2*(e^2*g - d*e*h + d^2*i)) + 9*c^2*e*(e*h - d*i)*x)
*Sqrt[1 - c^2*x^2])/(36*c^3*e^3) - (b*(e*h - d*i)*ArcSin[c*x])/(4*c^2*e^2) - ((I/2)*b*(e^3*f - d*e^2*g + d^2*e
*h - d^3*i)*ArcSin[c*x]^2)/e^4 + ((e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]))/e^3 + ((e*h - d*i)*x^2*(a + b
*ArcSin[c*x]))/(2*e^2) + (i*x^3*(a + b*ArcSin[c*x]))/(3*e) + (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x
]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 + (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*A
rcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e^3*f - d*e^2*g + d^2*e*h -
d^3*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x])
/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])
])/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2
])])/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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

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 2317

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4825

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

Rule 4837

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (f+g x+h x^2+109 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{d+e x} \, dx &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac {e x \left (654 d^2-3 d e (2 h+109 x)+e^2 \left (6 g+3 h x+218 x^2\right )\right )+\left (-654 d^3+6 e^3 f-6 d e^2 g+6 d^2 e h\right ) \log (d+e x)}{6 e^4 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {e x \left (654 d^2-3 d e (2 h+109 x)+e^2 \left (6 g+3 h x+218 x^2\right )\right )+\left (-654 d^3+6 e^3 f-6 d e^2 g+6 d^2 e h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{6 e^4}\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {e x \left (6 \left (109 d^2+e^2 g-d e h\right )-3 e (109 d-e h) x+218 e^2 x^2\right )}{\sqrt {1-c^2 x^2}}-\frac {6 \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{6 e^4}\\ &=\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {x \left (6 \left (109 d^2+e^2 g-d e h\right )-3 e (109 d-e h) x+218 e^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{6 e^3}+\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {b \int \frac {x \left (-2 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )+9 c^2 e (109 d-e h) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{18 c e^3}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(b (109 d-e h)) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c e^2}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (b c \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {\left (b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}+\frac {\left (b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {\left (i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}-\frac {\left (i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}\\ &=\frac {109 b x^2 \sqrt {1-c^2 x^2}}{9 c e}+\frac {b \left (4 \left (218 e^2+9 c^2 \left (109 d^2+e^2 g-d e h\right )\right )-9 c^2 e (109 d-e h) x\right ) \sqrt {1-c^2 x^2}}{36 c^3 e^3}+\frac {b (109 d-e h) \sin ^{-1}(c x)}{4 c^2 e^2}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x)^2}{2 e^4}+\frac {\left (109 d^2+e^2 g-d e h\right ) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}-\frac {(109 d-e h) x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 e^2}+\frac {109 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}-\frac {\left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {i b \left (109 d^3-e^3 f+d e^2 g-d^2 e h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1650\) vs. \(2(623)=1246\).
time = 4.96, size = 1650, normalized size = 2.65 \begin {gather*} \frac {144 a c^3 e \left (e^2 g-d e h+d^2 i\right ) x+72 a c^3 e^2 (e h-d i) x^2+48 a c^3 e^3 i x^3+144 a c^3 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \log (d+e x)+18 b c^2 e^2 g \left (8 e \sqrt {1-c^2 x^2}+8 c e x \text {ArcSin}(c x)-c d \left (i (\pi -2 \text {ArcSin}(c x))^2-32 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c d-e) \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )}{\sqrt {c^2 d^2-e^2}}\right )-4 \left (\pi +4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1-\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )-4 \left (\pi -4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1+\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+4 (\pi -2 \text {ArcSin}(c x)) \log (c (d+e x))+8 \text {ArcSin}(c x) \log (c (d+e x))+8 i \left (\text {PolyLog}\left (2,\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+\text {PolyLog}\left (2,-\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )\right )\right )\right )-72 i b c^3 e^3 f \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-18 b c e h \left (8 c d e \sqrt {1-c^2 x^2}+8 c^2 d e x \text {ArcSin}(c x)+4 i c^2 d^2 \text {ArcSin}(c x)^2+2 e^2 \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))-8 c^2 d^2 \text {ArcSin}(c x) \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-8 c^2 d^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+8 i c^2 d^2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+8 i c^2 d^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-e^2 \sin (2 \text {ArcSin}(c x))\right )+b i \left (144 c^2 d^2 e \sqrt {1-c^2 x^2}+36 e^3 \sqrt {1-c^2 x^2}+144 c^3 d^2 e x \text {ArcSin}(c x)+36 c e^3 x \text {ArcSin}(c x)-9 c d e^2 \left (i (\pi -2 \text {ArcSin}(c x))^2-32 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c d-e) \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )}{\sqrt {c^2 d^2-e^2}}\right )-4 \left (\pi +4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1-\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )-4 \left (\pi -4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1+\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+4 (\pi -2 \text {ArcSin}(c x)) \log (c (d+e x))+8 \text {ArcSin}(c x) \log (c (d+e x))+8 i \left (\text {PolyLog}\left (2,\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+\text {PolyLog}\left (2,-\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )\right )\right )+36 i c d \left (2 c^2 d^2-e^2\right ) \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+18 c d e^2 (2 \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))-\sin (2 \text {ArcSin}(c x)))-4 e^3 (\cos (3 \text {ArcSin}(c x))+3 \text {ArcSin}(c x) \sin (3 \text {ArcSin}(c x)))\right )}{144 c^3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x),x]

[Out]

(144*a*c^3*e*(e^2*g - d*e*h + d^2*i)*x + 72*a*c^3*e^2*(e*h - d*i)*x^2 + 48*a*c^3*e^3*i*x^3 + 144*a*c^3*(e^3*f
- d*e^2*g + d^2*e*h - d^3*i)*Log[d + e*x] + 18*b*c^2*e^2*g*(8*e*Sqrt[1 - c^2*x^2] + 8*c*e*x*ArcSin[c*x] - c*d*
(I*(Pi - 2*ArcSin[c*x])^2 - (32*I)*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]]*ArcTan[((c*d - e)*Cot[(Pi + 2*ArcSin[c*x]
)/4])/Sqrt[c^2*d^2 - e^2]] - 4*(Pi + 4*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]] - 2*ArcSin[c*x])*Log[1 - (I*(-(c*d) +
Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] - 4*(Pi - 4*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]] - 2*ArcSin[c*x])*L
og[1 + (I*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + 4*(Pi - 2*ArcSin[c*x])*Log[c*(d + e*x)] + 8*Ar
cSin[c*x]*Log[c*(d + e*x)] + (8*I)*(PolyLog[2, (I*(-(c*d) + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + Pol
yLog[2, ((-I)*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))]))) - (72*I)*b*c^3*e^3*f*(ArcSin[c*x]*(ArcSin
[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]
))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] +
2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - 18*b*c*e*h*(8*c*d*e*Sqrt[1 - c^2*x^2] + 8
*c^2*d*e*x*ArcSin[c*x] + (4*I)*c^2*d^2*ArcSin[c*x]^2 + 2*e^2*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 8*c^2*d^2*ArcSin
[c*x]*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - 8*c^2*d^2*ArcSin[c*x]*Log[1 - (I*e*E^(
I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + (8*I)*c^2*d^2*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + S
qrt[c^2*d^2 - e^2])] + (8*I)*c^2*d^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - e^2*Sin
[2*ArcSin[c*x]]) + b*i*(144*c^2*d^2*e*Sqrt[1 - c^2*x^2] + 36*e^3*Sqrt[1 - c^2*x^2] + 144*c^3*d^2*e*x*ArcSin[c*
x] + 36*c*e^3*x*ArcSin[c*x] - 9*c*d*e^2*(I*(Pi - 2*ArcSin[c*x])^2 - (32*I)*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]]*A
rcTan[((c*d - e)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[c^2*d^2 - e^2]] - 4*(Pi + 4*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2
]] - 2*ArcSin[c*x])*Log[1 - (I*(-(c*d) + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] - 4*(Pi - 4*ArcSin[Sqrt[
1 + (c*d)/e]/Sqrt[2]] - 2*ArcSin[c*x])*Log[1 + (I*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + 4*(Pi
- 2*ArcSin[c*x])*Log[c*(d + e*x)] + 8*ArcSin[c*x]*Log[c*(d + e*x)] + (8*I)*(PolyLog[2, (I*(-(c*d) + Sqrt[c^2*d
^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + PolyLog[2, ((-I)*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))])) +
(36*I)*c*d*(2*c^2*d^2 - e^2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt
[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, ((-I)*e*E^(I
*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^
2])]) + 18*c*d*e^2*(2*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - Sin[2*ArcSin[c*x]]) - 4*e^3*(Cos[3*ArcSin[c*x]] + 3*Arc
Sin[c*x]*Sin[3*ArcSin[c*x]])))/(144*c^3*e^4)

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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. 3469 vs. \(2 (621 ) = 1242\).
time = 1.26, size = 3470, normalized size = 5.57

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3470\)
default	\(\text {Expression too large to display}\)	\(3470\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/c*(-a*c/e^2*ln(c*e*x+c*d)*d*g+I*b*c/e*d^2*h/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^
2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+I*b*c/e*d^2*h/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/
2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*c/e*d^2*h*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c
*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-b*c/e*d^2*h*arcsin(c*x)/(c^2*d^2-e^
2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+b*c^3/e^3*d^4*h*
arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(
1/2)))+b*c^3/e^3*d^4*h*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/
(I*d*c+(-c^2*d^2+e^2)^(1/2)))-I*b*c^3/e^3*d^4*h/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*
d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+b/e^3*(-c^2*x^2+1)^(1/2)*d^2*i-1/36*b/c^2*i/e*cos(3*arcsin(c*x))
+1/4*b/c^2/e*(-c^2*x^2+1)^(1/2)*i-b*c^3/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1
/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c^3/e*d^2*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(
I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c^3/e*d^2*f*arcsin(c*x)/(c^2*d
^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*c^3/e^2
*d^3*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/
2)))-I*b*c^3/e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^
2+e^2)^(1/2)))*d^2-b*c^3/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+
e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-I*b*c^3/e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+
(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2+I*b*c^3/e^2*d^3*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+
(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-a*c/e^4*ln(c*e*x+c*d)*d^3*i+a/e^3*d^2*
i*c*x+1/3*a*c/e*i*x^3-1/8*b/c/e^2*sin(2*arcsin(c*x))*d*i-1/12*b/c^2*arcsin(c*x)*i/e*sin(3*arcsin(c*x))+b*arcsi
n(c*x)*g/e*c*x-1/2*I*b*c*arcsin(c*x)^2/e*f-I*b*c^3/e^3*d^4*h/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^
(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-1/2*a*c/e^2*d*i*x^2+1/4*b/c/e^2*arcsin(c*x)*cos(2*a
rcsin(c*x))*d*i+1/2*I*b*c*arcsin(c*x)^2/e^4*d^3*i+b*arcsin(c*x)/e^3*d^2*i*c*x+1/4*b/c/e*arcsin(c*x)*i*x+b*g/e*
(-c^2*x^2+1)^(1/2)+a*g/e*c*x+a*c/e*ln(c*e*x+c*d)*f+1/2*I*b*c*arcsin(c*x)^2/e^2*d*g+I*b*c*e*f/(c^2*d^2-e^2)*dil
og((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-b*c*e*f*arcsin(c*x)
/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*
c*e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2
)))+b*c*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^
2*d^2+e^2)^(1/2)))-b*c*e*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/
2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b*c*d*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+
e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1
/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b*c*d*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x
^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-1/2*I*b*c*arcsin(c*x)^2/e^3*d^2*h-b*arcsin(c*
x)/e^2*d*h*c*x-b*c^3/e^4*d^5*i*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)
^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+b*c/e^2*d^3*i*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^
(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*c^3/e^4*d^5*i*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c
+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b*c/e^2*d^3*i/(c^2*d^2-e^2
)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b*c/e^2*d^3*
i/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-
b/e^2*(-c^2*x^2+1)^(1/2)*d*h+1/8*b/c/e*h*sin(2*arcsin(c*x))+a*c/e^3*ln(c*e*x+c*d)*d^2*h-a/e^2*d*h*c*x+1/2*a*c/
e*h*x^2-1/4*b/c/e*h*arcsin(c*x)*cos(2*arcsin(c*x))+I*b*c^3/e^4*d^5*i/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2
*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c/e^2*d^3*i*arcsin(c*x)/(c^2*d^2-e^2)*ln(
(I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+I*b*c^3/e^4*d^5*i/(c^2
*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2))))

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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((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="maxima")

[Out]

a*f*e^(-1)*log(x*e + d) - (d*e^(-2)*log(x*e + d) - x*e^(-1))*a*g + 1/2*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2
*d*x)*e^(-2))*a*h - 1/6*I*(6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*a + I*b*integ
rate(x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(x*e + d), x) + integrate((b*h*x^2 + b*g*x + b*f)*arctan2(
c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(x*e + d), 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((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="fricas")

[Out]

integral(1/2*(2*a*h*x^2 + 2*I*a*x^3 + 2*a*g*x + 2*a*f + (-I*b*h*x^2 + b*x^3 - I*b*g*x - I*b*f)*log(-2*c^2*x^2
- 2*sqrt(c^2*x^2 - 1)*c*x + 1))/(x*e + d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d),x)

[Out]

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

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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((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="giac")

[Out]

integrate((i*x^3 + h*x^2 + g*x + f)*(b*arcsin(c*x) + a)/(e*x + d), 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 )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x),x)

[Out]

int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x), x)

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