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

Optimal. Leaf size=338 \[ -\frac{2 b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 a b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}+\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+2 b^2 \sqrt{c^2 d x^2+d}-\frac{2 b^2 c x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.342552, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5742, 5760, 4182, 2531, 2282, 6589, 5653, 261} \[ -\frac{2 b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{2 a b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}+\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+2 b^2 \sqrt{c^2 d x^2+d}-\frac{2 b^2 c x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5742

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 + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f
, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 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 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 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 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 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]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx &=\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\sqrt{d+c^2 d x^2} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b c \sqrt{d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{2 a b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c \sqrt{d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{2 a b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 c x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{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 (2 b \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 (2 b \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 (2 b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=2 b^2 \sqrt{d+c^2 d x^2}-\frac{2 a b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 c x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{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{2 b \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{2 b \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 (2 b^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 (2 b^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}}\\ &=2 b^2 \sqrt{d+c^2 d x^2}-\frac{2 a b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 c x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{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{2 b \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{2 b \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 (2 b^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 (2 b^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}}\\ &=2 b^2 \sqrt{d+c^2 d x^2}-\frac{2 a b c x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{2 b^2 c x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{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{2 b \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{2 b \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{2 b^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{2 b^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*}

Mathematica [A]  time = 1.16183, size = 352, normalized size = 1.04 \[ \frac{2 a b \sqrt{c^2 d x^2+d} \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 \sqrt{c^2 d x^2+d} \left (2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )+2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt{c^2 x^2+1}}+a^2 \sqrt{c^2 d x^2+d}-a^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+a^2 \sqrt{d} \log (c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.277, size = 823, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

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,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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}, x\right ) \end{align*}

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,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, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x}\, dx \end{align*}

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,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}

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,x, algorithm="giac")

[Out]

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