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

Optimal. Leaf size=180 \[ -b c^2 d \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 c^2 d \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right )-\frac{d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{b c d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac{c^2 d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+\frac{1}{2} c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2+c^2 d \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+b^2 c^2 d \log (x) \]

[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

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Rubi [A]  time = 0.310279, antiderivative size = 179, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5739, 5659, 3716, 2190, 2531, 2282, 6589, 5737, 29, 5675} \[ b c^2 d \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 c^2 d \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-\frac{d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{b c d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac{c^2 d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+\frac{1}{2} c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2+c^2 d \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+b^2 c^2 d \log (x) \]

Warning: Unable to verify antiderivative.

[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 5739

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*d^IntPart[p]*(d + e*x^2)^FracPart[p
])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[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 5659

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

Rule 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 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 29

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

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.366043, size = 212, normalized size = 1.18 \[ \frac{1}{2} d \left (2 a b c^2 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )-\frac{1}{3} b^2 c^2 \left (-6 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+2 \sinh ^{-1}(c x)^2 \left (\sinh ^{-1}(c x)-3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )\right )+2 a^2 c^2 \log (x)-\frac{a^2}{x^2}-\frac{2 a b \left (c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)\right )}{x^2}-\frac{b^2 \left (-2 c^2 x^2 \log (c x)+2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2\right )}{x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(-(a^2/x^2) - (2*a*b*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + 2*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 +
c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + 2*a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] + 2*
Log[1 - E^(-2*ArcSinh[c*x])]) - PolyLog[2, E^(-2*ArcSinh[c*x])]) - (b^2*c^2*(2*ArcSinh[c*x]^2*(ArcSinh[c*x] -
3*Log[1 - E^(2*ArcSinh[c*x])]) - 6*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 3*PolyLog[3, E^(2*ArcSinh[c*x
])]))/3))/2

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Maple [B]  time = 0.246, size = 515, normalized size = 2.9 \begin{align*}{c}^{2}d{a}^{2}\ln \left ( cx \right ) -{\frac{d{a}^{2}}{2\,{x}^{2}}}-{\frac{{c}^{2}d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{3}}-{\frac{cd{b}^{2}{\it Arcsinh} \left ( cx \right ) }{x}\sqrt{{c}^{2}{x}^{2}+1}}+{c}^{2}d{b}^{2}{\it Arcsinh} \left ( cx \right ) -{\frac{d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2\,{x}^{2}}}+{c}^{2}d{b}^{2}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1}-1 \right ) -2\,{c}^{2}d{b}^{2}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +{c}^{2}d{b}^{2}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +{c}^{2}d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{2}d{b}^{2}{\it Arcsinh} \left ( cx \right ){\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) -2\,{c}^{2}d{b}^{2}{\it polylog} \left ( 3,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +{c}^{2}d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{2}d{b}^{2}{\it Arcsinh} \left ( cx \right ){\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -2\,{c}^{2}d{b}^{2}{\it polylog} \left ( 3,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -{c}^{2}dab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}-{\frac{cdab}{x}\sqrt{{c}^{2}{x}^{2}+1}}+{c}^{2}dab-{\frac{dab{\it Arcsinh} \left ( cx \right ) }{{x}^{2}}}+2\,{c}^{2}dab{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{2}dab{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{2}dab{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{2}dab{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c^{2} d \log \left (x\right ) - a b d{\left (\frac{\sqrt{c^{2} x^{2} + 1} c}{x} + \frac{\operatorname{arsinh}\left (c x\right )}{x^{2}}\right )} - \frac{a^{2} d}{2 \, x^{2}} + \int \frac{b^{2} c^{2} d \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{x} + \frac{2 \, a b c^{2} d \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x} + \frac{b^{2} d \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} c^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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