3.7.45 \(\int \frac {a+b \text {ArcSin}(c x)}{x^3 (d+e x^2)^3} \, dx\) [645]

Optimal. Leaf size=783 \[ -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {ArcSin}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {ArcSin}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {ArcSin}(c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{d^4}-\frac {3 i b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{2 d^4} \]

[Out]

1/2*(-a-b*arcsin(c*x))/d^3/x^2-1/4*e*(a+b*arcsin(c*x))/d^2/(e*x^2+d)^2-e*(a+b*arcsin(c*x))/d^3/(e*x^2+d)+1/8*b
*c*e*(2*c^2*d+e)*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(3/2)-3*e*(a+b*arcsin(
c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4-3/2*I*b*e*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^4+3/2*
I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4
+b*c*e*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(1/2)-1/2*b*c*(-c^2*x^2+1)^(1/2)
/d^3/x+1/8*b*c*e^2*x*(-c^2*x^2+1)^(1/2)/d^3/(c^2*d+e)/(e*x^2+d)

________________________________________________________________________________________

Rubi [A]
time = 0.86, antiderivative size = 783, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4817, 4723, 270, 4721, 3798, 2221, 2317, 2438, 4813, 390, 385, 211, 4825, 4617} \begin {gather*} \frac {3 e (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^4}+\frac {3 e (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^4}+\frac {3 e (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^4}+\frac {3 e (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^4}-\frac {3 e \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^4}-\frac {e (a+b \text {ArcSin}(c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \text {ArcSin}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {ArcSin}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}+\frac {3 i b e \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 d^4}+\frac {b c e \left (2 c^2 d+e\right ) \text {ArcTan}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c e \text {ArcTan}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*c*Sqrt[1 - c^2*x^2])/(d^3*x) + (b*c*e^2*x*Sqrt[1 - c^2*x^2])/(8*d^3*(c^2*d + e)*(d + e*x^2)) - (a + b*
ArcSin[c*x])/(2*d^3*x^2) - (e*(a + b*ArcSin[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcSin[c*x]))/(d^3*(d + e
*x^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]) + (b*c*e*(2
*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)) + (3*e*(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^4) + (3*e*(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^4) + (3*e*(a + b*ArcSi
n[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(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^4) - (3*e*(a + b*ArcSin[c*x])*
Log[1 - E^((2*I)*ArcSin[c*x])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] -
Sqrt[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e
])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^4 - (
((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^4 + (((3*I)/2)*b*e*P
olyLog[2, E^((2*I)*ArcSin[c*x])])/d^4

Rule 211

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -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 3798

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

Rule 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 4721

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

Rule 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 4813

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

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^3 \left (d+e x^2\right )^3} \, dx &=\int \left (\frac {a+b \sin ^{-1}(c x)}{d^3 x^3}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right )}{d^4 x}+\frac {e^2 x \left (a+b \sin ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x \left (a+b \sin ^{-1}(c x)\right )}{d^4 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \sin ^{-1}(c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{d^2}\\ &=-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{2 d^3}-\frac {(3 e) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^4}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d^2}+\frac {\left (3 e^2\right ) \int \left (-\frac {a+b \sin ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^4}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {3 i e \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^4}+\frac {(6 i e) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{d^4}+\frac {(b c e) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{d^3}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d^3 \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {3 i e \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^4}+\frac {b c e \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^4}+\frac {(3 b e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^4}-\frac {\left (3 e^{3/2}\right ) \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^4}+\frac {\left (3 e^{3/2}\right ) \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^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{8 d^3 \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^4}-\frac {(3 i b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^4}-\frac {\left (3 i e^{3/2}\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^4}-\frac {\left (3 i e^{3/2}\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^4}+\frac {\left (3 i e^{3/2}\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^4}+\frac {\left (3 i e^{3/2}\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^4}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^4}+\frac {3 i b e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^4}-\frac {(3 b e) \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^4}-\frac {(3 b e) \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^4}-\frac {(3 b e) \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^4}-\frac {(3 b e) \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^4}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^4}+\frac {3 i b e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^4}+\frac {(3 i b e) \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^4}+\frac {(3 i b e) \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^4}+\frac {(3 i b e) \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^4}+\frac {(3 i b e) \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^4}\\ &=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}+\frac {3 e \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^4}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^4}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}-\frac {3 i b e \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^4}+\frac {3 i b e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^4}\\ \end {align*}

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Mathematica [F]
time = 6.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {ArcSin}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.68, size = 1820, normalized size = 2.32

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1820\)
default	\(\text {Expression too large to display}\)	\(1820\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*(5/4*I*b*(d*c^2*(c^2*d+e))^(1/2)/d^3/(c^2*d+e)^2*arctanh(1/4*(4*c^2*d-2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2+2*e
)/(c^4*d^2+c^2*d*e)^(1/2))*e-3*I*b/c^2/d^4*e^2/(c^2*d+e)*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-3/4*I*b/c^2/d^4*e^2/(
c^2*d+e)*sum((-_R1^2*e+4*c^2*d+e)/(-_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)+d
ilog((_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*I*b/c^2/d^4*e^2/(c^2*d+e
)*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+3/8*I*b*c^4/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e+3/4*I*b/c^2/d^4*e^3/(c^2*d+e
)*sum((_R1^2-1)/(-_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))-1/2*b*c^5/x/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(-
c^2*x^2+1)^(1/2)-1/2*b*c^4/x^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arcsin(c*x)-3*b/c^2/d^4*e^2/(c^2*d+e)*arcsin(c*x)
*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-9/4*b*c^2/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arcsin(c*x)*e^2-3/2*b*c^4/d^2/(c^2
*e*x^2+c^2*d)^2/(c^2*d+e)*arcsin(c*x)*e^2*x^2-b*c^5/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(-c^2*x^2+1)^(1/2)*e*x-1/2
*b*c^5/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(-c^2*x^2+1)^(1/2)*e^2*x^3-1/4*a*c^2*e/d^2/(c^2*e*x^2+c^2*d)^2+3/2*a/
c^2*e/d^4*ln(c^2*e*x^2+c^2*d)-3*a/c^2/d^4*e*ln(c*x)+1/2*I*b*c^6/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)+9/8*I*b/c^2*(d*c
^2*(c^2*d+e))^(1/2)/d^4/(c^2*d+e)^2*arctanh(1/4*(4*c^2*d-2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*
e)^(1/2))*e^2-3/2*b*c^2*x^2/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arcsin(c*x)*e^3-3/8*b*c^3*x^3/d^3/(c^2*e*x^2+c^2
*d)^2/(c^2*d+e)*(-c^2*x^2+1)^(1/2)*e^3+3/8*I*b*c^4*x^4/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e^3+3/4*I*b*c^4*x^2/d
^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e^2-3*I*b/d^3/(c^2*d+e)*e*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-3/4*I*b/d^3*e/(c^2*
d+e)*sum((-_R1^2*e+4*c^2*d+e)/(-_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*I*b/d^3*e/(c^2*d+e)*dilog(1+
I*c*x+(-c^2*x^2+1)^(1/2))+3/4*I*b/d^3*e^2/(c^2*d+e)*sum((_R1^2-1)/(-_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))-a*e/d^3/(c^2*e*x^2+c^2*d)-1/2*a/d^3/c^2/x^2-3*b/d^3/(c^2*d+e)*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2
))-9/4*b*c^4/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e*arcsin(c*x)-1/2*b*c^3/x/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(-c^2*x
^2+1)^(1/2)*e-7/8*b*c^3*x/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(-c^2*x^2+1)^(1/2)*e^2-1/2*b*c^2/x^2/d/(c^2*e*x^2+
c^2*d)^2/(c^2*d+e)*e*arcsin(c*x)+I*b*c^6*x^2/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e+1/2*I*b*c^6*x^4/d^2/(c^2*e*x^2+
c^2*d)^2/(c^2*d+e)*e^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^3/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/4*a*((6*x^4*e^2 + 9*d*x^2*e + 2*d^2)/(d^3*x^6*e^2 + 2*d^4*x^4*e + d^5*x^2) - 6*e*log(x^2*e + d)/d^4 + 12*e*
log(x)/d^4) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(x^9*e^3 + 3*d*x^7*e^2 + 3*d^2*x^5*e + d^
3*x^3), 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^3/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*arcsin(c*x) + a)/(x^9*e^3 + 3*d*x^7*e^2 + 3*d^2*x^5*e + d^3*x^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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^3\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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