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

Optimal. Leaf size=213 \[ \frac{b \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{c^2 x^2+1}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}+\frac{\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{b^2 \log \left (c^2 x^2+1\right )}{2 c^4 d^2} \]

[Out]

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

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Rubi [A]  time = 0.404791, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5751, 5714, 3718, 2190, 2531, 2282, 6589, 5675, 260} \[ \frac{b \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{c^2 x^2+1}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}+\frac{\log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{b^2 \log \left (c^2 x^2+1\right )}{2 c^4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5751

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p
+ 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*
x^2)^FracPart[p])/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Ar
cSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 1]

Rule 5714

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

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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 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 260

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

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx &=-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac{b \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{c d^2}+\frac{\int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{c^2 d}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d^2}+\frac{b \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{c^3 d^2}+\frac{b^2 \int \frac{x}{1+c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d^2}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d^2}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 d^2}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ &=-\frac{b x \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d^2 \sqrt{1+c^2 x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c^4 d^2}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}+\frac{b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b^2 \text{Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ \end{align*}

Mathematica [C]  time = 1.06315, size = 320, normalized size = 1.5 \[ \frac{4 a b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )+4 a b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+2 b^2 \left (-\sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} \log \left (c^2 x^2+1\right )+\frac{\sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-\frac{c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )\right )+\frac{a^2}{c^2 x^2+1}+a^2 \log \left (c^2 x^2+1\right )-\frac{a b \left (\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right )}{c x+i}-\frac{a b \left (\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right )}{c x-i}-a b \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-a b \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )}{2 c^4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.197, size = 499, normalized size = 2.3 \begin{align*}{\frac{{a}^{2}}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{a}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{d}^{2}{c}^{4}}}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{3\,{d}^{2}{c}^{4}}}-{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ) x}{{c}^{3}{d}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ){x}^{2}}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }}-2\,{\frac{{b}^{2}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{2}{c}^{4}}}+{\frac{{b}^{2}}{{d}^{2}{c}^{4}}\ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{d}^{2}{c}^{4}}\ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }+{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-{\frac{{b}^{2}}{2\,{d}^{2}{c}^{4}}{\it polylog} \left ( 3,- \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-{\frac{ab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{d}^{2}{c}^{4}}}-{\frac{abx}{{c}^{3}{d}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{ab{x}^{2}}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{ab{\it Arcsinh} \left ( cx \right ) }{{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{ab}{{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{ab{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{{d}^{2}{c}^{4}}}+{\frac{ab}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c^4*a^2/d^2/(c^2*x^2+1)+1/2/c^4*a^2/d^2*ln(c^2*x^2+1)-1/3/c^4*b^2/d^2*arcsinh(c*x)^3-1/c^3*b^2/d^2*arcsinh
(c*x)/(c^2*x^2+1)^(1/2)*x+1/c^2*b^2/d^2*arcsinh(c*x)/(c^2*x^2+1)*x^2+1/2/c^4*b^2/d^2*arcsinh(c*x)^2/(c^2*x^2+1
)+1/c^4*b^2/d^2*arcsinh(c*x)/(c^2*x^2+1)-2/c^4*b^2/d^2*ln(c*x+(c^2*x^2+1)^(1/2))+1/c^4*b^2/d^2*ln(1+(c*x+(c^2*
x^2+1)^(1/2))^2)+1/c^4*b^2/d^2*arcsinh(c*x)^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+1/c^4*b^2/d^2*arcsinh(c*x)*polyl
og(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/c^4/d^2-1/c^4*a*b/d^2*arcsinh(c
*x)^2-1/c^3*a*b/d^2/(c^2*x^2+1)^(1/2)*x+1/c^2*a*b/d^2*x^2/(c^2*x^2+1)+1/c^4*a*b/d^2*arcsinh(c*x)/(c^2*x^2+1)+1
/c^4*a*b/d^2/(c^2*x^2+1)+2/c^4*a*b/d^2*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+1/c^4*a*b/d^2*polylog(2,-(
c*x+(c^2*x^2+1)^(1/2))^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{1}{c^{6} d^{2} x^{2} + c^{4} d^{2}} + \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4} d^{2}}\right )} + \frac{{\left (b^{2} +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right )\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{2 \,{\left (c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}} - \int -\frac{{\left (2 \, a b c^{4} x^{4} - b^{2} c^{2} x^{2} - b^{2} -{\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) +{\left (2 \, a b c^{3} x^{3} - b^{2} c x -{\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \log \left (c^{2} x^{2} + 1\right )\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{8} d^{2} x^{5} + 2 \, c^{6} d^{2} x^{3} + c^{4} d^{2} x +{\left (c^{7} d^{2} x^{4} + 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*(1/(c^6*d^2*x^2 + c^4*d^2) + log(c^2*x^2 + 1)/(c^4*d^2)) + 1/2*(b^2 + (b^2*c^2*x^2 + b^2)*log(c^2*x^2
+ 1))*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^2*x^2 + c^4*d^2) - integrate(-(2*a*b*c^4*x^4 - b^2*c^2*x^2 - b^2 -
(b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) + (2*a*b*c^3*x^3 - b^2*c*x - (b^2*c^3*x^3 + b^2*c*x)*log
(c^2*x^2 + 1))*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c^8*d^2*x^5 + 2*c^6*d^2*x^3 + c^4*d^2*x + (c^7
*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2)*sqrt(c^2*x^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{3} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{3}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3)/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2} x^{3}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b x^{3} \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2*x**3/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b**2*x**3*asinh(c*x)**2/(c**4*x**4 + 2*c**2*x
**2 + 1), x) + Integral(2*a*b*x**3*asinh(c*x)/(c**4*x**4 + 2*c**2*x**2 + 1), x))/d**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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