\(\int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx\) [71]

Optimal result

Integrand size = 12, antiderivative size = 205 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

[Out]

-1/3*arcsinh(b*x+a)^3+arcsinh(b*x+a)^2*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))+arcsinh(b*x+a)^2*ln
(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+2*arcsinh(b*x+a)*polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a
^2+1)^(1/2)))+2*arcsinh(b*x+a)*polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-2*polylog(3,(b*x+a+(1+
(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))-2*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5859, 5827, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{3} \text {arcsinh}(a+b x)^3 \]

[In]

Int[ArcSinh[a + b*x]^2/x,x]

[Out]

-1/3*ArcSinh[a + b*x]^3 + ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + ArcSinh[a + b*x
]^2*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] + 2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sq
rt[1 + a^2])] + 2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh
[a + b*x]/(a - Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 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 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {1}{3} \text {arcsinh}(a+b x)^3+\frac {\text {Subst}\left (\int \frac {e^x x^2}{-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{b}+\frac {\text {Subst}\left (\int \frac {e^x x^2}{-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 \text {Subst}\left (\int x \log \left (1+\frac {e^x}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )-2 \text {Subst}\left (\int x \log \left (1+\frac {e^x}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^x}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )-2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^x}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\text {arcsinh}(a+b x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {x}{a-\sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )-2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {x}{a+\sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.22 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a+b x)^3+\text {arcsinh}(a+b x)^2 \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\text {arcsinh}(a+b x)^2 \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+2 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

[In]

Integrate[ArcSinh[a + b*x]^2/x,x]

[Out]

-1/3*ArcSinh[a + b*x]^3 + ArcSinh[a + b*x]^2*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b)] + ArcS
inh[a + b*x]^2*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b)] + 2*ArcSinh[a + b*x]*PolyLog[2, -(E^
ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b))] + 2*ArcSinh[a + b*x]*PolyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b
) + Sqrt[1 + a^2]/b)*b))] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh[a +
b*x]/(a + Sqrt[1 + a^2])]

Maple [F]

\[\int \frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{x}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integral(arcsinh(b*x + a)^2/x, x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(asinh(a + b*x)**2/x, x)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(b*x + a)^2/x, x)

Giac [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(b*x + a)^2/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^2}{x} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x} \,d x \]

[In]

int(asinh(a + b*x)^2/x,x)

[Out]

int(asinh(a + b*x)^2/x, x)