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

Optimal. Leaf size=166 \[ -b d \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 d \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{1}{2} b c d x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+d \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^2 d x^2 \]

[Out]

(b^2*c^2*d*x^2)/4 - (b*c*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 - (d*(a + b*ArcSinh[c*x])^2)/4 + (d*(1
+ c^2*x^2)*(a + b*ArcSinh[c*x])^2)/2 + (d*(a + b*ArcSinh[c*x])^3)/(3*b) + d*(a + b*ArcSinh[c*x])^2*Log[1 - E^(
-2*ArcSinh[c*x])] - b*d*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (b^2*d*PolyLog[3, E^(-2*ArcSinh
[c*x])])/2

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Rubi [A]  time = 0.246114, antiderivative size = 165, 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 = {5744, 5659, 3716, 2190, 2531, 2282, 6589, 5682, 5675, 30} \[ b d \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 d \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{1}{2} b c d x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+d \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^2 d x^2 \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b^2*c^2*d*x^2)/4 - (b*c*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 - (d*(a + b*ArcSinh[c*x])^2)/4 + (d*(1
+ c^2*x^2)*(a + b*ArcSinh[c*x])^2)/2 - (d*(a + b*ArcSinh[c*x])^3)/(3*b) + d*(a + b*ArcSinh[c*x])^2*Log[1 - E^(
2*ArcSinh[c*x])] + b*d*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])] - (b^2*d*PolyLog[3, E^(2*ArcSinh[c*
x])])/2

Rule 5744

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int
[(f*x)^m*(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 + 2*p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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 5682

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

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
]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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} \, dx &=\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-(b c d) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+d \operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{2} (b c d) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{2} \left (b^2 c^2 d\right ) \int x \, dx\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-(2 d) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-(2 b d) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (b^2 d\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b 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 d\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b 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 d \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.373892, size = 209, normalized size = 1.26 \[ \frac{1}{8} d \left (-8 a b \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+8 b^2 \left (\sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )+4 a^2 c^2 x^2+8 a^2 \log (x)-4 a b \left (c x \sqrt{c^2 x^2+1}-\sinh ^{-1}(c x)\right )+8 a b c^2 x^2 \sinh ^{-1}(c x)+8 a b \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-2 b^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )+b^2 \left (2 \sinh ^{-1}(c x)^2+1\right ) \cosh \left (2 \sinh ^{-1}(c x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(4*a^2*c^2*x^2 - 4*a*b*(c*x*Sqrt[1 + c^2*x^2] - ArcSinh[c*x]) + 8*a*b*c^2*x^2*ArcSinh[c*x] + b^2*(1 + 2*Arc
Sinh[c*x]^2)*Cosh[2*ArcSinh[c*x]] + 8*a*b*ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 - E^(-2*ArcSinh[c*x])]) + 8*a^2
*Log[x] - 8*a*b*PolyLog[2, E^(-2*ArcSinh[c*x])] + 8*b^2*(-ArcSinh[c*x]^3/3 + ArcSinh[c*x]^2*Log[1 - E^(2*ArcSi
nh[c*x])] + ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - PolyLog[3, E^(2*ArcSinh[c*x])]/2) - 2*b^2*ArcSinh[c*
x]*Sinh[2*ArcSinh[c*x]]))/8

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Maple [B]  time = 0.125, size = 425, normalized size = 2.6 \begin{align*}{\frac{d{a}^{2}{c}^{2}{x}^{2}}{2}}+d{a}^{2}\ln \left ( cx \right ) -{\frac{d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{3}}+{\frac{d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2}}{2}}-{\frac{d{b}^{2}{\it Arcsinh} \left ( cx \right ) cx}{2}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{4}}+{\frac{{b}^{2}{c}^{2}d{x}^{2}}{4}}+{\frac{d{b}^{2}}{8}}+d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,d{b}^{2}{\it Arcsinh} \left ( cx \right ){\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) -2\,d{b}^{2}{\it polylog} \left ( 3,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +d{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,d{b}^{2}{\it Arcsinh} \left ( cx \right ){\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -2\,d{b}^{2}{\it polylog} \left ( 3,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) -dab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}+dab{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-{\frac{dabcx}{2}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{dab{\it Arcsinh} \left ( cx \right ) }{2}}+2\,dab{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,dab{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,dab{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +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,x)

[Out]

1/2*d*a^2*c^2*x^2+d*a^2*ln(c*x)-1/3*d*b^2*arcsinh(c*x)^3+1/2*d*b^2*arcsinh(c*x)^2*c^2*x^2-1/2*d*b^2*arcsinh(c*
x)*(c^2*x^2+1)^(1/2)*c*x+1/4*d*b^2*arcsinh(c*x)^2+1/4*b^2*c^2*d*x^2+1/8*d*b^2+d*b^2*arcsinh(c*x)^2*ln(1+c*x+(c
^2*x^2+1)^(1/2))+2*d*b^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-2*d*b^2*polylog(3,-c*x-(c^2*x^2+1)^(1/
2))+d*b^2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))+2*d*b^2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))-2*d
*b^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))-d*a*b*arcsinh(c*x)^2+d*a*b*arcsinh(c*x)*c^2*x^2-1/2*d*a*b*c*x*(c^2*x^2+1
)^(1/2)+1/2*d*a*b*arcsinh(c*x)+2*d*a*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+2*d*a*b*polylog(2,-c*x-(c^2*x^
2+1)^(1/2))+2*d*a*b*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+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*} \frac{1}{2} \, a^{2} c^{2} d x^{2} + a^{2} d \log \left (x\right ) + \int b^{2} c^{2} d x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, a b c^{2} d x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \frac{b^{2} d \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{x} + \frac{2 \, a b d \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x}\,{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,x, algorithm="maxima")

[Out]

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

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

[Out]

d*(Integral(a**2/x, x) + Integral(a**2*c**2*x, x) + Integral(b**2*asinh(c*x)**2/x, x) + Integral(2*a*b*asinh(c
*x)/x, x) + Integral(b**2*c**2*x*asinh(c*x)**2, x) + Integral(2*a*b*c**2*x*asinh(c*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}\,{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,x, algorithm="giac")

[Out]

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