3.7.12 \(\int \frac {a+b \sinh ^{-1}(c x)}{(d+e x^2)^2} \, dx\) [612]

Optimal. Leaf size=707 \[ -\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \text {ArcTan}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \text {ArcTan}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]

[Out]

-1/4*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/
2)+1/4*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(
1/2)-1/4*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e
^(1/2)+1/4*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)
/e^(1/2)+1/4*b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-
1/4*b*polylog(2,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)+1/4*b*poly
log(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4*b*polylog(2,(c*
x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4*b*c*arctan((-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)^(1/2)/e^(1/2)-1/4*b*c*arctan((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)^(1/2)/e^(1/2)+1/4*(-a-b*arcsinh(c*x))/d/e^(1/2)/((-d)^(1
/2)-x*e^(1/2))+1/4*(a+b*arcsinh(c*x))/d/e^(1/2)/((-d)^(1/2)+x*e^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 0.78, antiderivative size = 707, 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 = {5793, 5828, 739, 210, 5827, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \text {ArcTan}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}-\frac {b c \text {ArcTan}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a + b*ArcSinh[c*x])/(d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + S
qrt[e]*x)) - (b*c*ArcTan[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]
*Sqrt[e]) - (b*c*ArcTan[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*
Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^
(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4
*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]
)])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d
) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))
])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)
^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*
Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 5680

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

Rule 5793

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

Rule 5827

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

Rule 5828

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*
((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[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 \sinh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-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 \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {\int \frac {a+b \sinh ^{-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 \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}-\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}+\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{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^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-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 {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.21, size = 622, normalized size = 0.88 \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 (-2 \sqrt {d} \left (-\frac {\sinh ^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \text {ArcTan}\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 )+2 i \sqrt {d} \left (\frac {\sinh ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {c \tanh ^{-1}\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 )+i \left (\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )-i \left (\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )}{4 d^{3/2} \sqrt {e}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[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*(-2*Sqrt[d]*(-(ArcSinh[c*x]/(I
*Sqrt[d] + Sqrt[e]*x)) + (c*ArcTan[(Sqrt[e] - I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*
d - e]) + (2*I)*Sqrt[d]*(ArcSinh[c*x]/(Sqrt[d] + I*Sqrt[e]*x) + (c*ArcTanh[(I*Sqrt[e] - c^2*Sqrt[d]*x)/(Sqrt[c
^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d - e]) + I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSi
nh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) + e])] + Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) +
e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + 2*PolyLog[2, -((Sqrt[
e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))]) - I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e
]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqr
t[-(c^2*d) + e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))] + 2*Poly
Log[2, (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])))/(4*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 = 22.51, size = 1766, normalized size = 2.50

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1766\)
default	\(\text {Expression too large to display}\)	\(1766\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(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*arcsinh(c*x)*x/d/
(c^2*e*x^2+c^2*d)+1/4*b*c^2/d*sum(1/_R1/(_R1^2*e+2*c^2*d-e)*(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+
dilog((_R1-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)*(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)),_R1=RootO
f(e*_Z^4+(4*c^2*d-2*e)*_Z^2+e))+b*c^6*(-(2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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)/e^3+b*c^4*(-(2*c^2*d-2*(c^2*d*(c^2*d-
e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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^2*d*(c^2*d-e))^(1/2)-b*c^4*(-(2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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^2-1/2*b*c^2*(-(2*c^2*d-2*(c^2*d*(c^
2*d-e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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)/e^2*(c^2*d*(c^2*d-e))^(1/2)-b*c^4*(-(2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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-b*c^2*(-(2*c^2*d-2*(c^2*d*(c^2*d-e))^
(1/2)-e)*e)^(1/2)*arctanh(e*(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^3*(c
^2*d*(c^2*d-e))^(1/2)+1/2*b*c^2*(-(2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh(e*(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^2+b*c^6*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(
1/2)*arctan(e*(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)/e^3-b*c^4*(
(2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan(e*(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^2*d*(c^2*d-e))^(1/2)-b*c^4*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)
*arctan(e*(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^2+1/2*b*c^2*((2
*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan(e*(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)/e^2*(c^2*d*(c^2*d-e))^(1/2)-b*c^4*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)
*arctan(e*(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+b*c^2*((2*c^2*d+2*(c^2*
d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan(e*(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^3*(c^2*d*(c^2*d-e))^(1/2)+1/2*b*c^2*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan(e*(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^2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*d-2.718281828459045>0)', s
ee `assume?`

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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*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*arcsinh(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 {asinh}{\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*asinh(c*x))/(e*x**2+d)**2,x)

[Out]

Integral((a + b*asinh(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*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arcsinh(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 {asinh}\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*asinh(c*x))/(d + e*x^2)^2,x)

[Out]

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

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