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

Optimal. Leaf size=485 \[ -\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]
) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqr
t[e]) + ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]
*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt
[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))])/(2*Sqrt[-d]*Sqrt
[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*Po
lyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2,
(Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e])

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Rubi [A]  time = 0.832494, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5706, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]
) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqr
t[e]) + ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]
*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt
[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))])/(2*Sqrt[-d]*Sqrt
[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*Po
lyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2,
(Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e])

Rule 5706

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 5799

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 5561

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 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]

Rubi steps

\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{c \sqrt{-d}-\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{c \sqrt{-d}+\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{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 )}{2 \sqrt{-d}}-\frac{\operatorname{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 )}{2 \sqrt{-d}}-\frac{\operatorname{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 )}{2 \sqrt{-d}}-\frac{\operatorname{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 )}{2 \sqrt{-d}}\\ &=\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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{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 )}{2 \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \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 )}{2 \sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.426022, size = 434, normalized size = 0.89 \[ \frac{b \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )-b \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right )-b \sqrt{d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )+b \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )+2 a \sqrt{-d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )-b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )+b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right )+b \sqrt{d} \sinh ^{-1}(c x) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )-b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d^2} \sqrt{e}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.424, size = 224, normalized size = 0.5 \begin{align*}{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{bc}{2}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{1}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e+2\,{c}^{2}d-e \right ) } \left ({\it Arcsinh} \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}+{\frac{bc}{2}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{\it \_R1}}{{{\it \_R1}}^{2}e+2\,{c}^{2}d-e} \left ({\it Arcsinh} \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*c*b*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))+1/2*c*b*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=RootOf(e*_Z^4+(4*c^2*d-2*e)*_Z^2+e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arsinh}\left (c x\right ) + a}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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