3.264 \(\int \frac {\sqrt {d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2}{x^3} \, dx\)

Optimal. Leaf size=358 \[ -\frac {b c^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {b c^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.38, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5737, 5661, 266, 63, 208, 5760, 4182, 2531, 2282, 6589} \[ -\frac {b c^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {b c^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

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 208

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

Rule 266

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 2282

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 2531

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 4182

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 5661

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 5737

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*Sqrt[d + e*x^2])/(f*(m +
1)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d + e*x^2])/(f
^2*(m + 1)*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 5760

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

Rule 6589

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*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x^2} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 4.88, size = 446, normalized size = 1.25 \[ \frac {1}{8} \left (-\frac {4 a^2 \sqrt {c^2 d x^2+d}}{x^2}-4 a^2 c^2 \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+d\right )+4 a^2 c^2 \sqrt {d} \log (x)+\frac {2 a b c^2 \sqrt {c^2 d x^2+d} \left (4 \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-4 \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {b^2 c^2 \sqrt {c^2 d x^2+d} \left (8 \sinh ^{-1}(c x) \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-8 \sinh ^{-1}(c x) \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+8 \text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )-8 \text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+4 \sinh ^{-1}(c x) \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+\sinh ^{-1}(c x)^2 \left (-\text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-\sinh ^{-1}(c x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+8 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{\sqrt {c^2 x^2+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^3,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*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*Ar
cSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[
c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/Sqrt
[1 + c^2*x^2] + (b^2*c^2*Sqrt[d + c^2*d*x^2]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSi
nh[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*L
og[Tanh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSi
nh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] - 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c
*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2])/8

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.45, size = 870, normalized size = 2.43 \[ -\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {a^{2} \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right ) c^{2}}{2}+\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2}}{2}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2}}{2 \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}\, c}{x \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \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}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \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}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arctanh \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c^{2}}{c^{2} x^{2}+1}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c}{x \sqrt {c^{2} x^{2}+1}}-\frac {a b \arcsinh \left (c x \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{x^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (c^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \sqrt {c^{2} d x^{2} + d} c^{2} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{2}}\right )} a^{2} + \int \frac {\sqrt {c^{2} d x^{2} + d} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x^{3}} + \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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