3.7.39 \(\int \frac {a+b \text {ArcSin}(c x)}{(d+e x^2)^2} \, dx\) [639]

Optimal. Leaf size=757 \[ -\frac {a+b \text {ArcSin}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {ArcSin}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c \tanh ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}-\frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {ArcSin}(c x)) \log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {ArcSin}(c x)) \log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 757, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4757, 4827, 739, 212, 4825, 4617, 2221, 2317, 2438} \begin {gather*} -\frac {(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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(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 )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(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 )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \text {ArcSin}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {ArcSin}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \text {Li}_2\left (-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \tanh ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c \tanh ^{-1}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

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

Rule 739

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4617

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b
^2, 2] + b*E^(I*(c + d*x)))), x], x] + Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E
^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 4757

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

Rule 4825

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

Rule 4827

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

Rubi steps

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

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Mathematica [A]
time = 1.14, size = 591, normalized size = 0.78 \begin {gather*} \frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (i \sqrt {d} \left (\frac {\text {ArcSin}(c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \text {ArcTan}\left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+\sqrt {d} \left (\frac {\text {ArcSin}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+i \text {ArcSin}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-i \text {ArcSin}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+\text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-\text {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {e} e^{i \text {ArcSin}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{2 d^{3/2} \sqrt {e}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*(I*Sqrt[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
]) + Sqrt[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]*Sq
rt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + I*ArcSin[c*x]*(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])]) - I*ArcSin[c*x]*(Log[1 + (S
qrt[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] + Sq
rt[c^2*d + e])]) + PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] - PolyLog[2, (Sqrt[e]
*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] +
Sqrt[c^2*d + e]))] + PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(2*d^(3/2)*Sqrt[e
]))/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.22, size = 1706, normalized size = 2.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1706\)
default	\(\text {Expression too large to display}\)	\(1706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(1/2*a*c^3*x/d/(c^2*e*x^2+c^2*d)+1/2*a*c/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*b*c^3*arcsin(c*x)*x/d/(
c^2*e*x^2+c^2*d)-b*c^6*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)*d+b*c^4*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)*(d*
c^2*(c^2*d+e))^(1/2)-b*c^4*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)/e^2+1/2*b*c^2*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/(c^2*d+e)
/e^2*(d*c^2*(c^2*d+e))^(1/2)+b*c^4*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3-b*c^2*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/e^3*(d*c^2*(c^2
*d+e))^(1/2)+1/2*b*c^2*((2*c^2*d+2*(d*c^2*(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*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/e^2-b*c^6*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1/2)*ar
ctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)*d-b*c^4*(-e*
(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e)
)^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)*(d*c^2*(c^2*d+e))^(1/2)-b*c^4*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1
/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)/e^2-1/2*b*
c^2*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(
c^2*d+e))^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)/e^2*(d*c^2*(c^2*d+e))^(1/2)+b*c^4*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1
/2)+e))^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3+b*c^2*
(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*
d+e))^(1/2)-e)*e)^(1/2))/d/e^3*(d*c^2*(c^2*d+e))^(1/2)+1/2*b*c^2*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e))^(1
/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/e^2-1/4*b*c^2/d*su
m(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))-1/4*b*c^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)))

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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))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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