3.3.35 \(\int \frac {x^3 (a+b \sinh ^{-1}(c x))^2}{(d+c^2 d x^2)^2} \, dx\) [235]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {5810, 5797, 3799, 2221, 2611, 2320, 6724, 5783, 266} \begin {gather*} \frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\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 {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 {b^2 \text {Li}_3\left (-e^{2 \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} \end {gather*}

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 266

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]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3799

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 5783

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

Rule 5797

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 5810

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/(2*c*(p + 1)))*Simp[
(d + e*x^2)^p/(1 + c^2*x^2)^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] && LtQ[p, -1] && IGtQ[m, 1]

Rule 6724

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 {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 {\text {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 \text {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) \text {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 \text {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 \text {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] Result contains complex when optimal does not.
time = 0.61, size = 320, normalized size = 1.50 \begin {gather*} \frac {\frac {a^2}{1+c^2 x^2}-\frac {a b \left (\sqrt {1+c^2 x^2}-i \sinh ^{-1}(c x)\right )}{i+c x}-\frac {a b \left (\sqrt {1+c^2 x^2}+i \sinh ^{-1}(c x)\right )}{-i+c x}-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 )+a^2 \log \left (1+c^2 x^2\right )+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 (-\frac {c x \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\sinh ^{-1}(c x)^2}{2+2 c^2 x^2}+\frac {1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )+\frac {1}{2} \log \left (1+c^2 x^2\right )-\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 )\right )}{2 c^4 d^2} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

Maple [A]
time = 7.23, size = 454, normalized size = 2.13

method	result	size
derivativedivides	\(\frac {\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 d^{2}}-\frac {b^{2} \arcsinh \left (c x \right ) c x}{d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \arcsinh \left (c x \right ) c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {2 b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}-\frac {a b \arcsinh \left (c x \right )^{2}}{d^{2}}-\frac {a b c x}{d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {a b \,c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}}{c^{4}}\)	\(454\)
default	\(\frac {\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 d^{2}}-\frac {b^{2} \arcsinh \left (c x \right ) c x}{d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \arcsinh \left (c x \right ) c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {2 b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}-\frac {a b \arcsinh \left (c x \right )^{2}}{d^{2}}-\frac {a b c x}{d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {a b \,c^{2} x^{2}}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b \arcsinh \left (c x \right )}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a b}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}}{c^{4}}\)	\(454\)

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,method=_RETURNVERBOSE)

[Out]

1/c^4*(1/2*a^2/d^2/(c^2*x^2+1)+1/2*a^2/d^2*ln(c^2*x^2+1)-1/3*b^2/d^2*arcsinh(c*x)^3-b^2/d^2*arcsinh(c*x)/(c^2*
x^2+1)^(1/2)*c*x+b^2/d^2*arcsinh(c*x)/(c^2*x^2+1)*c^2*x^2+1/2*b^2/d^2*arcsinh(c*x)^2/(c^2*x^2+1)+b^2/d^2*arcsi
nh(c*x)/(c^2*x^2+1)+b^2/d^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-2*b^2/d^2*ln(c*x+(c^2*x^2+1)^(1/2))+b^2/d^2*arcsin
h(c*x)^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+b^2/d^2*arcsinh(c*x)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*po
lylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^2-a*b/d^2*arcsinh(c*x)^2-a*b/d^2/(c^2*x^2+1)^(1/2)*c*x+a*b/d^2/(c^2*x^2+
1)*c^2*x^2+a*b/d^2*arcsinh(c*x)/(c^2*x^2+1)+a*b/d^2/(c^2*x^2+1)+2*a*b/d^2*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(
1/2))^2)+a*b/d^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________