\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \sqrt {d+c^2 d x^2}} \, dx\) [298]

Optimal result

Integrand size = 28, antiderivative size = 360 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \sqrt {d+c^2 d x^2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 d x^2}+\frac {c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {d+c^2 d x^2}}+\frac {b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}+\frac {b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5809, 5816, 4267, 2611, 2320, 6724, 5776, 272, 65, 214} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\frac {c^2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}+\frac {b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {b c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 d x^2}-\frac {b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 d x^2+d}}+\frac {b^2 c^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \sqrt {c^2 x^2+1} \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 4267

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5809

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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5816

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 3.81 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\frac {-\frac {4 a^2 \sqrt {d+c^2 d x^2}}{x^2}-4 a^2 c^2 \sqrt {d} \log (x)+4 a^2 c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {2 a b c^2 d^2 \left (1+c^2 x^2\right )^{3/2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}+\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right )^{3/2} \left (-4 \text {arcsinh}(c x) \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-8 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )+8 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}}{8 d} \]

[In]

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

[Out]

((-4*a^2*Sqrt[d + c^2*d*x^2])/x^2 - 4*a^2*c^2*Sqrt[d]*Log[x] + 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*
d*x^2]] + (2*a*b*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 -
4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcS
inh[c*x])] + 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/
(d + c^2*d*x^2)^(3/2) + (b^2*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^
2*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[
c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] - 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] + 8*ArcSinh[c*x]*PolyLog[
2, E^(-ArcSinh[c*x])] - 8*PolyLog[3, -E^(-ArcSinh[c*x])] + 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Se
ch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(d + c^2*d*x^2)^(3/2))/(8*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(779\) vs. \(2(373)=746\).

Time = 0.31 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.17

method	result	size
default	\(-\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{2 d \,x^{2}}+\frac {a^{2} c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b^{2} \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \operatorname {arcsinh}\left (c x \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (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 ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (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 ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(780\)
parts	\(-\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{2 d \,x^{2}}+\frac {a^{2} c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b^{2} \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \operatorname {arcsinh}\left (c x \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (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 ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (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 ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (-\frac {\left (\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(780\)

[In]

int((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*a^2/d/x^2*(c^2*d*x^2+d)^(1/2)+1/2*a^2*c^2/d^(1/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)+b^2*(-1/2*(ar
csinh(c*x)*c^2*x^2+2*c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x))*arcsinh(c*x)*(d*(c^2*x^2+1))^(1/2)/x^2/d/(c^2*x^2+1)+
1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^2+(d*(c^2*x^2+1))^(
1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c^2-(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^
(1/2)/d*polylog(3,-c*x-(c^2*x^2+1)^(1/2))*c^2-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)^2*ln(
1-c*x-(c^2*x^2+1)^(1/2))*c^2-(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^
(1/2))*c^2+(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polylog(3,c*x+(c^2*x^2+1)^(1/2))*c^2-2*(d*(c^2*x^2+1))^(1
/2)/(c^2*x^2+1)^(1/2)/d*arctanh(c*x+(c^2*x^2+1)^(1/2))*c^2)+2*a*b*(-1/2*(arcsinh(c*x)*c^2*x^2+c*x*(c^2*x^2+1)^
(1/2)+arcsinh(c*x))*(d*(c^2*x^2+1))^(1/2)/x^2/d/(c^2*x^2+1)-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcs
inh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c^2-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polylog(2,c*x+(c^2*x^2+
1)^(1/2))*c^2+1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^2+1/2*(
d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c^2)

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^2*d*x^5 + d*x^3), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate((a+b*asinh(c*x))**2/x**3/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asinh(c*x))**2/(x**3*sqrt(d*(c**2*x**2 + 1))), x)

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^(1/2)), x)