3.7.11 \(\int \frac {a+b \sinh ^{-1}(c x)}{d+e x^2} \, dx\) [611]

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

[Out]

1/2*(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)^(1/2)/e^(1/2
)-1/2*(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)^(1/2)/e^(1
/2)+1/2*(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)^(1/2)/e^
(1/2)-1/2*(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)^(1/2)/
e^(1/2)-1/2*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)^(1/2)/e^(1/2)+1
/2*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)^(1/2)/e^(1/2)-1/2*b*polyl
og(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*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)^(1/2)/e^(1/2)

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Rubi [A]
time = 0.59, antiderivative size = 485, normalized size of antiderivative = 1.00, 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 = {5793, 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 )}{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}}-\frac {b \text {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \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 )}{2 \sqrt {-d} \sqrt {e}} \end {gather*}

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 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]

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 {\text {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 {\text {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 {\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 )}{2 \sqrt {-d}}-\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 )}{2 \sqrt {-d}}-\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 )}{2 \sqrt {-d}}-\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 )}{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 \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 )}{2 \sqrt {-d} \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 )}{2 \sqrt {-d} \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 )}{2 \sqrt {-d} \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 )}{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 {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 \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 )}{2 \sqrt {-d} \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 )}{2 \sqrt {-d} \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 )}{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*}

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Mathematica [A]
time = 0.31, size = 434, normalized size = 0.89 \begin {gather*} \frac {2 a \sqrt {-d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \sinh ^{-1}(c x) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 11.79, size = 233, normalized size = 0.48

method	result	size
derivativedivides	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}}{c}\)	\(233\)
default	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}}{c}\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a*c/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*b*c^2*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*
b*c^2*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]
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*arcsinh(c*x))/(e*x^2+d),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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