\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) [205]

Optimal result

Integrand size = 24, antiderivative size = 180 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5807, 5775, 3797, 2221, 2611, 2320, 6724, 5805, 29, 5783} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-b c^2 d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {b c d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2+c^2 d \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x) \]

[In]

Int[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^3,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5805

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

Rule 5807

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1
+ c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

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 {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+(b c d) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx+\left (c^2 d\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x} \, dx \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {\left (c^2 d\right ) \text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}+\left (b^2 c^2 d\right ) \int \frac {1}{x} \, dx+\left (b c^3 d\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+b^2 c^2 d \log (x)+\frac {\left (2 c^2 d\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-\left (2 c^2 d\right ) \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+\left (b c^2 d\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \left (b^2 c^2 d\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {a^2 d}{2 x^2}+2 a b c^2 d \left (-\frac {\sqrt {1+c^2 x^2}}{2 c x}-\frac {\text {arcsinh}(c x)}{2 c^2 x^2}\right )+a^2 c^2 d \log (x)+b^2 c^2 d \left (-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{2 c^2 x^2}+\log (c x)\right )+2 a b c^2 d \left (-\frac {1}{2} \text {arcsinh}(c x)^2+\text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+b^2 c^2 d \left (-\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right ) \]

[In]

Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(197)=394\).

Time = 0.20 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.24

method	result	size
derivativedivides	\(c^{2} \left (d \,a^{2} \left (\ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\)	\(404\)
default	\(c^{2} \left (d \,a^{2} \left (\ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\)	\(404\)
parts	\(d \,a^{2} \left (-\frac {1}{2 x^{2}}+c^{2} \ln \left (x \right )\right )+d \,b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(405\)

[In]

int((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="fricas")

[Out]

integral((a^2*c^2*d*x^2 + a^2*d + (b^2*c^2*d*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*c^2*d*x^2 + a*b*d)*arcsinh(c
*x))/x^3, x)

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=d \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {a^{2} c^{2}}{x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \]

[In]

integrate((c**2*d*x**2+d)*(a+b*asinh(c*x))**2/x**3,x)

[Out]

d*(Integral(a**2/x**3, x) + Integral(a**2*c**2/x, x) + Integral(b**2*asinh(c*x)**2/x**3, x) + Integral(2*a*b*a
sinh(c*x)/x**3, x) + Integral(b**2*c**2*asinh(c*x)**2/x, x) + Integral(2*a*b*c**2*asinh(c*x)/x, x))

Maxima [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxima")

[Out]

a^2*c^2*d*log(x) - a*b*d*(sqrt(c^2*x^2 + 1)*c/x + arcsinh(c*x)/x^2) - 1/2*a^2*d/x^2 + integrate(b^2*c^2*d*log(
c*x + sqrt(c^2*x^2 + 1))^2/x + 2*a*b*c^2*d*log(c*x + sqrt(c^2*x^2 + 1))/x + b^2*d*log(c*x + sqrt(c^2*x^2 + 1))
^2/x^3, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x^3} \,d x \]

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2))/x^3,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2))/x^3, x)