3.3.5 \(\int \frac {(d+c^2 d x^2) (a+b \sinh ^{-1}(c x))^2}{x^3} \, dx\) [205]

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

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 180, normalized size of antiderivative = 1.00, 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 = {5807, 5775, 3797, 2221, 2611, 2320, 6724, 5805, 29, 5783} \begin {gather*} -b c^2 d \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \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-\frac {1}{2} b^2 c^2 d \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 c^2 d \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 29

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

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 3797

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

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5805

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/(f*(m + 1)))*Simp[Sqrt[d
+ e*x^2]/Sqrt[1 + c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x] - Dist[(c^2/(f^2*(m + 1))
)*Simp[Sqrt[d + e*x^2]/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 5807

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/(f*(m + 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] && GtQ[p, 0] && LtQ[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 {\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.21, size = 212, normalized size = 1.18 \begin {gather*} \frac {1}{2} d \left (-\frac {a^2}{x^2}-\frac {2 a b \left (c x \sqrt {1+c^2 x^2}+\sinh ^{-1}(c x)\right )}{x^2}+2 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}+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 (2 \sinh ^{-1}(c x)^2 \left (\sinh ^{-1}(c x)-3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )-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 )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(477\) vs. \(2(197)=394\).
time = 6.48, size = 478, normalized size = 2.66

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a^{2} d}{2 c^{2} x^{2}}+a^{2} d \ln \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{3}}{3}-\frac {b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}+b^{2} d \arcsinh \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}+b^{2} d \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-b d a \arcsinh \left (c x \right )^{2}-\frac {b d a \sqrt {c^{2} x^{2}+1}}{c x}+b d a -\frac {b d a \arcsinh \left (c x \right )}{c^{2} x^{2}}+2 b d a \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(478\)
default	\(c^{2} \left (-\frac {a^{2} d}{2 c^{2} x^{2}}+a^{2} d \ln \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{3}}{3}-\frac {b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{c x}+b^{2} d \arcsinh \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{2 c^{2} x^{2}}+b^{2} d \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-b d a \arcsinh \left (c x \right )^{2}-\frac {b d a \sqrt {c^{2} x^{2}+1}}{c x}+b d a -\frac {b d a \arcsinh \left (c x \right )}{c^{2} x^{2}}+2 b d a \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(478\)

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,method=_RETURNVERBOSE)

[Out]

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

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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((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 = 0.00, 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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