3.648 \(\int \frac {a+b \sin ^{-1}(c x)}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=1092 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 \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 {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]

[Out]

1/16*b*c^3*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/(c^2*d+e)^(3/2)/e^(1/2)+1
/16*b*c^3*arctanh((c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/(c^2*d+e)^(3/2)/e^(1/2)+3/1
6*(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)/e^(1/
2)-3/16*(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)
/e^(1/2)+3/16*(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)/e^(1/2)-3/16*(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)/e^(1/2)+3/16*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)))/
(-d)^(5/2)/e^(1/2)+3/16*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)))/(-
d)^(5/2)/e^(1/2)-3/16*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)))/(-d)^
(5/2)/e^(1/2)-3/16*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)))/(-d)^(5/
2)/e^(1/2)+1/16*(-a-b*arcsin(c*x))/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)-x*e^(1/2))^2-3/16*(a+b*arcsin(c*x))/d^2/e^(1
/2)/((-d)^(1/2)-x*e^(1/2))+1/16*(a+b*arcsin(c*x))/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)+x*e^(1/2))^2+3/16*(a+b*arcsin
(c*x))/d^2/e^(1/2)/((-d)^(1/2)+x*e^(1/2))+3/16*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))/d^2/e^(1/2)/(c^2*d+e)^(1/2)+3/16*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))/d^2/e^(1/2)/(c^2*d+e)^(1/2)+1/16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)-x*e^(1/2))
+1/16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)+x*e^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 1.25, antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4667, 4743, 731, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 \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 {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^
(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^
2) - (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt
[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTa
nh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*
ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (b
*c^3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2))
+ (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d
+ e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)
^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])]
)/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^
2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d]
+ Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sq
rt[-d] - Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*
Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I
*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/
(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e])

Rule 206

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

Rule 725

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 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 4521

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 4667

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 4741

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 4743

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*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac {e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {3 e \left (a+b \sin ^{-1}(c x)\right )}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {(3 e) \int \frac {a+b \sin ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 d^2}-\frac {(3 e) \int \frac {a+b \sin ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 d^2}-\frac {(3 e) \int \frac {a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 d^2}-\frac {e^{3/2} \int \frac {a+b \sin ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^3} \, dx}{8 (-d)^{3/2}}-\frac {e^{3/2} \int \frac {a+b \sin ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^3} \, dx}{8 (-d)^{3/2}}\\ &=-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d^2}-\frac {(3 b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d^2}+\frac {\left (b c \sqrt {e}\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 (-d)^{3/2}}-\frac {\left (b c \sqrt {e}\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2 \sqrt {1-c^2 x^2}} \, dx}{16 (-d)^{3/2}}-\frac {(3 e) \int \left (-\frac {\sqrt {-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{8 d^2}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{16 (-d)^{5/2}}-\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{16 (-d)^{5/2}}-\frac {(3 b c) \operatorname {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 )}{16 d^2}+\frac {(3 b c) \operatorname {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 )}{16 d^2}+\frac {\left (b c^3\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d \left (c^2 d+e\right )}-\frac {\left (b c^3\right ) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1-c^2 x^2}} \, dx}{16 d \left (c^2 d+e\right )}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}-\frac {3 \operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac {\left (b c^3\right ) \operatorname {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 )}{16 d \left (c^2 d+e\right )}+\frac {\left (b c^3\right ) \operatorname {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 )}{16 d \left (c^2 d+e\right )}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}-\frac {(3 i) \operatorname {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 )}{16 (-d)^{5/2}}-\frac {(3 i) \operatorname {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 )}{16 (-d)^{5/2}}-\frac {(3 i) \operatorname {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 )}{16 (-d)^{5/2}}-\frac {(3 i) \operatorname {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 )}{16 (-d)^{5/2}}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 i b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 i b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 i b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 i b) \operatorname {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 )}{16 (-d)^{5/2} \sqrt {e}}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c^3 \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}\\ \end {align*}

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Mathematica [A]  time = 6.10, size = 1055, normalized size = 0.97 \[ \frac {3 a x}{8 d^2 \left (e x^2+d\right )}+\frac {a x}{4 d \left (e x^2+d\right )^2}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+b \left (\frac {3 i \left (\frac {\sin ^{-1}(c x)}{i \sqrt {e} x+\sqrt {d}}-\frac {c \tan ^{-1}\left (\frac {\sqrt {d} x c^2+i \sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d c^2+e}}\right )}{16 d^2 \sqrt {e}}-\frac {3 \left (-\frac {\sin ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}-\frac {c \tanh ^{-1}\left (\frac {i \sqrt {d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d c^2+e}}\right )}{16 d^2 \sqrt {e}}+\frac {i \left (-\frac {i \sqrt {d} \left (\log \left (\frac {e \sqrt {d c^2+e} \left (-i \sqrt {d} x c^2+\sqrt {e}+\sqrt {d c^2+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {e} x \sqrt {d}\right )}\right )+\log (4)\right ) c^3}{\sqrt {e} \left (d c^2+e\right )^{3/2}}-\frac {\sqrt {1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt {e} x-i \sqrt {d}\right )}-\frac {\sin ^{-1}(c x)}{\sqrt {e} \left (\sqrt {e} x-i \sqrt {d}\right )^2}\right )}{16 d^{3/2}}-\frac {i \left (\frac {i \sqrt {d} \left (\log \left (\frac {e \sqrt {d c^2+e} \left (i \sqrt {d} x c^2+\sqrt {e}+\sqrt {d c^2+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )+\log (4)\right ) c^3}{\sqrt {e} \left (d c^2+e\right )^{3/2}}-\frac {\sqrt {1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt {e} x+i \sqrt {d}\right )}-\frac {\sin ^{-1}(c x)}{\sqrt {e} \left (\sqrt {e} x+i \sqrt {d}\right )^2}\right )}{16 d^{3/2}}-\frac {3 \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{c \sqrt {d}-\sqrt {d c^2+e}}+1\right )+\log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{\sqrt {d} c+\sqrt {d c^2+e}}+1\right )\right )\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d c^2+e}-c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )\right )}{32 d^{5/2} \sqrt {e}}+\frac {3 \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac {e^{i \sin ^{-1}(c x)} \sqrt {e}}{\sqrt {d c^2+e}-c \sqrt {d}}+1\right )+\log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )\right )\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {d c^2+e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )\right )}{32 d^{5/2} \sqrt {e}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]
) + b*((((3*I)/16)*(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]))/(d^2*Sqrt[e]) - (3*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcT
anh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(16*d^2*Sqrt[e]) + ((I
/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] +
Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*
Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*(-((c*S
qrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (
I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])
)/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(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]))]))/(32*d^(5/2)*Sqrt[e]) + (3*(ArcS
in[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*Sqr
t[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(32*d^(5/
2)*Sqrt[e]))

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arcsin(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.12, size = 3110, normalized size = 2.85 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^7*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d
*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)^2*d+c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh
(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)^2*(c^2*d*(c^2*d+e
))^(1/2)-c^5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*(c^2*d*(c^2*d+e))^(1/2)+3/8*c*b*((2*c^2*d+2*(c^2*d*(c^
2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)
)/e/d^2/(c^2*d+e)-c^7*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/(
(-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*d+5/4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/
2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^2/d/(c^
2*d+e)+5/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2
*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^2/d/(c^2*d+e)-3/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)
^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)^2/e+3
/8*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*
d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e/d^2/(c^2*d+e)-3/4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arct
anh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/(c^2*d+e)^2/e+1/4*c^4*a*x/
d/(c^2*e*x^2+c^2*d)^2+3/16*c^3*b/d/(c^2*d+e)*sum(1/_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*c
^3*b/d/(c^2*d+e)*sum(_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))+1/8*c^5*b/(c^2*d+e)/(c^2*e*x^2+c^2*d
)^2*(-c^2*x^2+1)^(1/2)+3/8*c^2*a/d^2*x/(c^2*e*x^2+c^2*d)+3/8*a/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+3/16*c*
b/d^2/(c^2*d+e)*e*sum(1/_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*c*b/d^2/(c^2*d+e)*e*sum(_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))-7/4*c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arc
tanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/e^2-7/4*c^5*b*(
-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d
+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/e^2+c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x
+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)/e^3+c^5*b*(-(2*c^2*d-2*(c^2*d*
(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1
/2))/(c^2*d+e)/e^3+5/8*c^6*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x+3/8*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))
^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d^2/(
c^2*d+e)^2/e*(c^2*d*(c^2*d+e))^(1/2)-3/4*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+
(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^2/d^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)
-5/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(
c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)+c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d
+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e
^3/d/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-3/8*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*
x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d^2/(c^2*d+e)^2/e*(c^2*d*(c^2*d+e))^(1
/2)+3/4*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*
(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^2/d^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/8*c^5*b/d/(c^2*d+e)/(c^2*e*x^
2+c^2*d)^2*(-c^2*x^2+1)^(1/2)*x^2*e+3/8*c^4*b/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^3*e^2+5/8*c^4*b/
d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x*e+3/8*c^6*b*e/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^3+5/
4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*
d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)-c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^
(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/d/
(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, a {\left (\frac {3 \, e x^{3} + 5 \, d x}{d^{2} e^{2} x^{4} + 2 \, d^{3} e x^{2} + d^{4}} + \frac {3 \, \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{2}}\right )} + b \int \frac {\arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*((3*e*x^3 + 5*d*x)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) + 3*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^2)) + b*inte
grate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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