\(\int \frac {a+b \arcsin (c x)}{x^2 (d+e x^2)^2} \, dx\) [640]

Optimal result

Integrand size = 21, antiderivative size = 795 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a+b \arcsin (c x)}{d^2 x}+\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {c^2 d+e}}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {c^2 d+e}}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}} \]

[Out]

(-a-b*arcsin(c*x))/d^2/x-b*c*arctanh((-c^2*x^2+1)^(1/2))/d^2-3/4*(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)))*e^(1/2)/(-d)^(5/2)+3/4*(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)))*e^(1/2)/(-d)^(5/2)-3/4*(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)))*e^(1/2)/(-d)^(5/2)+3/4*(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)))*e^(1/2)/(-d)^(5/2)-3/4*I*b*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)))*e^(1/2)/(-d)^(5/2)+3/4*I*b*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)))*e^(1/2)/(-d)^(5/2)-3/4*I*b*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)))*e^(1/2)/(-d)^(5/2)+3/4*I*b*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)))*e^(1/2)/(-d)^(5/2)+1/4*(a+b*arcsin(c*x))*e^(1/2)/d^2/((-d)^(1/
2)-x*e^(1/2))-1/4*(a+b*arcsin(c*x))*e^(1/2)/d^2/((-d)^(1/2)+x*e^(1/2))-1/4*b*c*arctanh((-c^2*x*(-d)^(1/2)+e^(1
/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))*e^(1/2)/d^2/(c^2*d+e)^(1/2)-1/4*b*c*arctanh((c^2*x*(-d)^(1/2)+e^(1/2)
)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))*e^(1/2)/d^2/(c^2*d+e)^(1/2)

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4817, 4723, 272, 65, 214, 4757, 4827, 739, 212, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}-\frac {a+b \arcsin (c x)}{d^2 x}+\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {d c^2+e}}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {d c^2+e}}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}} \]

[In]

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

[Out]

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

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 212

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

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 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, 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 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]

Rule 4827

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

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

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 672, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {-\frac {8 a \sqrt {d}}{x}-\frac {4 a \sqrt {d} e x}{d+e x^2}-12 a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (-2 i \sqrt {d} \sqrt {e} \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 \sqrt {d} \sqrt {e} \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-\frac {8 \sqrt {d} \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{x}+3 \sqrt {e} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-3 \sqrt {e} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{8 d^{5/2}} \]

[In]

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

[Out]

((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((-2*I)*Sqrt[
d]*Sqrt[e]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[
1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 2*Sqrt[d]*Sqrt[e]*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqr
t[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - (8*Sqrt[d]*(ArcSin[c*x] + c*x
*ArcTanh[Sqrt[1 - c^2*x^2]]))/x + 3*Sqrt[e]*(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*ArcSi
n[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - 3*Sqrt[e]*(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])])))/(8*d^(5/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.11 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.17

method	result	size
parts	\(a \left (-\frac {1}{d^{2} x}-\frac {e \left (\frac {x}{2 e \,x^{2}+2 d}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{d^{2}}\right )+b c \left (-\frac {\arcsin \left (c x \right ) \left (3 c^{2} e \,x^{2}+2 c^{2} d \right )}{2 c x \,d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {3 e \left (\munderset {\textit {\_R1} =\operatorname {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 )+\operatorname {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 )}{16 d^{3} c^{2}}-\frac {3 e \left (\munderset {\textit {\_R1} =\operatorname {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 )+\operatorname {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 )}{16 d^{3} c^{2}}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 d^{2} e^{2}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 d^{2} e^{2}}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right )}\, \left (2 d^{2} c^{4}+2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 c^{2} e d +\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 d^{2} \left (c^{2} d +e \right ) e^{2}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, d \,c^{2}+2 d^{2} c^{4}+2 c^{2} e d -\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {d \,c^{2} \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 d^{2} \left (c^{2} d +e \right ) e^{2}}\right )\)	\(928\)
derivativedivides	\(\text {Expression too large to display}\)	\(962\)
default	\(\text {Expression too large to display}\)	\(962\)

[In]

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

[Out]

a*(-1/d^2/x-e/d^2*(1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))))+b*c*(-1/2/c/x*arcsin(c*x)*(3*c^2*
e*x^2+2*c^2*d)/d^2/(c^2*e*x^2+c^2*d)-1/d^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/d^2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)
+3/16/d^3*e/c^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))-3/16/d^3*e/c^2*su
m((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)+dilo
g((_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))-1/2*(-e*(2*c^2*d-2*(d*c^2*(c^
2*d+e))^(1/2)+e))^(1/2)*(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2
*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d^2/e^2-1/2*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(
d*c^2*(c^2*d+e))^(1/2)+e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)
)/d^2/e^2+1/2*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1/2)*(2*d^2*c^4+2*(d*c^2*(c^2*d+e))^(1/2)*d*c^2+2*c^
2*e*d+(d*c^2*(c^2*d+e))^(1/2)*e)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e
)^(1/2))/d^2/(c^2*d+e)/e^2+1/2*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(d*c^2*(c^2*d+e))^(1/2)*d*c
^2+2*d^2*c^4+2*c^2*e*d-(d*c^2*(c^2*d+e))^(1/2)*e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(d*c^2*(c^2
*d+e))^(1/2)+e)*e)^(1/2))/d^2/(c^2*d+e)/e^2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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