3.637 \(\int \frac {x^4 (a+b \sin ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=787 \[ \frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {a x}{e^2}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {b c d \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \tanh ^{-1}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \sin ^{-1}(c x)}{e^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.03, antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4733, 4619, 261, 4667, 4743, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ \frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \sin ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {a x}{e^2}+\frac {b c d \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \tanh ^{-1}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \sin ^{-1}(c x)}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 261

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

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 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 4619

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

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 4733

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

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Mathematica [A]  time = 1.56, size = 649, normalized size = 0.82 \[ \frac {\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 a \sqrt {e} x+b \left (3 \sqrt {d} \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d c^2+e}-c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )\right )\right )\right )-3 \sqrt {d} \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{c \sqrt {d}-\sqrt {d c^2+e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {d} c+\sqrt {d c^2+e}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}-c \sqrt {d}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d+e}+c \sqrt {d}}\right )\right )\right )\right )+2 i d \left (\frac {\sin ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \tan ^{-1}\left (\frac {c^2 \sqrt {d} x+i \sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{\sqrt {c^2 d+e}}\right )+2 d \left (\frac {c \tanh ^{-1}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{\sqrt {c^2 d+e}}+\frac {\sin ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}\right )+\frac {8 \sqrt {e} \left (\sqrt {1-c^2 x^2}+c x \sin ^{-1}(c x)\right )}{c}\right )}{8 e^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \arcsin \left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.08, size = 1738, normalized size = 2.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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