3.7.32 \(\int \frac {a+b \text {ArcSin}(c x)}{x^4 (d+e x^2)} \, dx\) [632]

Optimal. Leaf size=649 \[ -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \text {ArcSin}(c x)}{3 d x^3}+\frac {e (a+b \text {ArcSin}(c x))}{d^2 x}-\frac {b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}} \]

[Out]

1/3*(-a-b*arcsin(c*x))/d/x^3+e*(a+b*arcsin(c*x))/d^2/x-1/6*b*c^3*arctanh((-c^2*x^2+1)^(1/2))/d+b*c*e*arctanh((
-c^2*x^2+1)^(1/2))/d^2+1/2*e^(3/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(
c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*e^(3/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1
/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)+1/2*e^(3/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-
d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*e^(3/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I
*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)+1/2*I*b*e^(3/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*
(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*I*b*e^(3/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)
^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)+1/2*I*b*e^(3/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1
/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*I*b*e^(3/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+
(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/6*b*c*(-c^2*x^2+1)^(1/2)/d/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4817, 4723, 272, 44, 65, 214, 4757, 4825, 4617, 2221, 2317, 2438} \begin {gather*} \frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e (a+b \text {ArcSin}(c x))}{d^2 x}-\frac {a+b \text {ArcSin}(c x)}{3 d x^3}+\frac {i b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {b c e \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)),x]

[Out]

-1/6*(b*c*Sqrt[1 - c^2*x^2])/(d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e*(a + b*ArcSin[c*x]))/(d^2*x) - (b*c^
3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcSin[c*x])*Lo
g[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c
*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*A
rcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(
a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I
/2)*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)
*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(
3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2
)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(-d)^(5/2)

Rule 44

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

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

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 4617

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] + (Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b
^2, 2] + b*E^(I*(c + d*x)))), x], x] + Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E
^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

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 4757

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4817

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

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]

Rubi steps

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

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Mathematica [A]
time = 0.25, size = 531, normalized size = 0.82 \begin {gather*} -\frac {a}{3 d x^3}+\frac {a e}{d^2 x}+\frac {a e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+b \left (-\frac {e \left (-\frac {\text {ArcSin}(c x)}{x}-c \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\right )}{d^2}-\frac {c x \sqrt {1-c^2 x^2}+2 \text {ArcSin}(c x)+c^3 x^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{6 d x^3}-\frac {e^{3/2} \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}+\frac {e^{3/2} \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)),x]

[Out]

-1/3*a/(d*x^3) + (a*e)/(d^2*x) + (a*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) + b*(-((e*(-(ArcSin[c*x]/x) -
c*ArcTanh[Sqrt[1 - c^2*x^2]]))/d^2) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x
^2]])/(6*d*x^3) - (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] -
Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt
[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[
d] + Sqrt[c^2*d + e]))]))/(4*d^(5/2)) + (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcS
in[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]
)])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*Ar
cSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(4*d^(5/2)))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.35, size = 488, normalized size = 0.75

method	result	size
derivativedivides	\(c^{3} \left (\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}-\frac {b \sqrt {-c^{2} x^{2}+1}}{6 d \,c^{2} x^{2}}-\frac {b \arcsin \left (c x \right )}{3 d \,c^{3} x^{3}}+\frac {b \arcsin \left (c x \right ) e}{c^{3} d^{2} x}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c^{4} d^{3}}+\frac {b \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6 d}-\frac {b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6 d}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c^{4} d^{3}}-\frac {b e \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{c^{2} d^{2}}+\frac {b e \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} d^{2}}\right )\)	\(488\)
default	\(c^{3} \left (\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}-\frac {b \sqrt {-c^{2} x^{2}+1}}{6 d \,c^{2} x^{2}}-\frac {b \arcsin \left (c x \right )}{3 d \,c^{3} x^{3}}+\frac {b \arcsin \left (c x \right ) e}{c^{3} d^{2} x}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c^{4} d^{3}}+\frac {b \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6 d}-\frac {b \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6 d}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c^{4} d^{3}}-\frac {b e \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{c^{2} d^{2}}+\frac {b e \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} d^{2}}\right )\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/x^4/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

c^3*(a/c^3*e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/3*a/d/c^3/x^3+a/c^3/d^2*e/x-1/6*b/d/c^2/x^2*(-c^2*x^2
+1)^(1/2)-1/3*b/d*arcsin(c*x)/c^3/x^3+b/c^3*arcsin(c*x)/d^2*e/x-1/8*b/c^4/d^3*e^2*sum((-_R1^2*e+4*c^2*d+e)/_R1
/(-_R1^2*e+2*c^2*d+e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2
))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))+1/6*b/d*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-1/6*b/d*ln(1+I*c*x+(
-c^2*x^2+1)^(1/2))-1/8*b/c^4/d^3*e^2*sum((4*_R1^2*c^2*d+_R1^2*e-e)/_R1/(-_R1^2*e+2*c^2*d+e)*(I*arcsin(c*x)*ln(
(_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e
)*_Z^2+e))-b/c^2/d^2*e*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+b/c^2/d^2*e*ln(1+I*c*x+(-c^2*x^2+1)^(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((a+b*arcsin(c*x))/x^4/(e*x^2+d),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(e*x^2+d),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))/x**4/(e*x**2+d),x)

[Out]

Integral((a + b*asin(c*x))/(x**4*(d + e*x**2)), 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((a+b*arcsin(c*x))/x^4/(e*x^2+d),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))/(x^4*(d + e*x^2)),x)

[Out]

int((a + b*asin(c*x))/(x^4*(d + e*x^2)), x)

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