3.3.71 \(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))^2}{x^2} \, dx\) [271]

Optimal. Leaf size=398 \[ \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]

[Out]

-(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x+1/4*b^2*c^2*d*x*(c^2*d*x^2+d)^(1/2)+3/2*c^2*d*x*(a+b*arcsinh(c*x))
^2*(c^2*d*x^2+d)^(1/2)-5/4*b^2*c*d*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/2*b*c^3*d*x^2*(a+b*arc
sinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+c*d*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2
)+1/2*c*d*(a+b*arcsinh(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/(c^2*x^2+1)^(1/2)+2*b*c*d*(a+b*arcsinh(c*x))*ln(1-1/(c*x+
(c^2*x^2+1)^(1/2))^2)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-b^2*c*d*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^
2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+b*c*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5807, 5785, 5783, 5776, 327, 221, 5801, 5775, 3797, 2221, 2317, 2438, 201} \begin {gather*} \frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {c^2 x^2+1}}+\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b c^3 d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}-\frac {b^2 c d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 c^2 d x \sqrt {c^2 d x^2+d}-\frac {5 b^2 c d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*c^2*d*x*Sqrt[d + c^2*d*x^2])/4 - (5*b^2*c*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(4*Sqrt[1 + c^2*x^2]) - (3*
b*c^3*d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + b*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d +
c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (c*d*Sqrt[d + c^
2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x + (c*d*S
qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*Sqrt[1 + c^2*x^2]) + (2*b*c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSi
nh[c*x])])/Sqrt[1 + c^2*x^2]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 2317

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 5785

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[(1/2)*Simp[Sqrt[d + e*x^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/2)*Simp[Sqrt[d + e*x^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 5801

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

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]

Rubi steps

\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (3 c^2 d\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {1}{2} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{2 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}-\frac {\left (4 b c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c d \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 2.06, size = 369, normalized size = 0.93 \begin {gather*} \frac {12 a^2 d \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+24 a b d \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+c x \sinh ^{-1}(c x)^2+2 c x \log (c x)\right )+36 a^2 c d^{3/2} x \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-8 b^2 d \sqrt {d+c^2 d x^2} \left (\sinh ^{-1}(c x) \left (3 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-c x \sinh ^{-1}(c x) \left (3+\sinh ^{-1}(c x)\right )-6 c x \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+3 c x \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )+b^2 c d x \sqrt {d+c^2 d x^2} \left (4 \sinh ^{-1}(c x)^3-6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )+\left (3+6 \sinh ^{-1}(c x)^2\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )-6 a b c d x \sqrt {d+c^2 d x^2} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{24 x \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a^2*d*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 24*a*b*d*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2
*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) + 36*a^2*c*d^(3/2)*x*Sqrt[1 + c^2*x^2]*Log[c*d*x + S
qrt[d]*Sqrt[d + c^2*d*x^2]] - 8*b^2*d*Sqrt[d + c^2*d*x^2]*(ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - c*
x*ArcSinh[c*x]*(3 + ArcSinh[c*x]) - 6*c*x*Log[1 - E^(-2*ArcSinh[c*x])]) + 3*c*x*PolyLog[2, E^(-2*ArcSinh[c*x])
]) + b^2*c*d*x*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + (3 + 6*ArcSinh[c*
x]^2)*Sinh[2*ArcSinh[c*x]]) - 6*a*b*c*d*x*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[
c*x] + Sinh[2*ArcSinh[c*x]])))/(24*x*Sqrt[1 + c^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs. \(2(372)=744\).
time = 2.45, size = 954, normalized size = 2.40

method	result	size
default	\(-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d \arcsinh \left (c x \right ) x}{c^{2} x^{2}+1}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \arcsinh \left (c x \right ) x^{3}}{c^{2} x^{2}+1}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \arcsinh \left (c x \right )^{2} x^{3}}{2 c^{2} x^{2}+2}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d \,x^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c d}{\sqrt {c^{2} x^{2}+1}}+\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c d}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c d \arcsinh \left (c x \right )}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) d}{x \left (c^{2} x^{2}+1\right )}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {3 a^{2} c^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{2}-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c d}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c d}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c d \arcsinh \left (c x \right )}{4 \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c d}{\sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c d}{4 \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d}{x \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3} c d}{2 \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c d}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c d}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} d \arcsinh \left (c x \right ) x^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d \arcsinh \left (c x \right )^{2} x}{2 \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{4} d \,x^{3}}{4 c^{2} x^{2}+4}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} d x}{4 c^{2} x^{2}+4}\)	\(954\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*b*(d*(c^2*x^2+1))^(1/2)*c^2*d/(c^2*x^2+1)*arcsinh(c*x)*x+3/2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ar
csinh(c*x)^2*c*d-1/2*a*b*(d*(c^2*x^2+1))^(1/2)*c^3*d/(c^2*x^2+1)^(1/2)*x^2-2*a*b*(d*(c^2*x^2+1))^(1/2)*c*d/(c^
2*x^2+1)^(1/2)*arcsinh(c*x)-2*a*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)*d/x/(c^2*x^2+1)+2*a*b*(d*(c^2*x^2+1))^(1/
2)/(c^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*c*d+3/2*a^2*c^2*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d
)^(1/2))/(c^2*d)^(1/2)+3/2*a^2*c^2*d*x*(c^2*d*x^2+d)^(1/2)-a^2/d/x*(c^2*d*x^2+d)^(5/2)+a^2*c^2*x*(c^2*d*x^2+d)
^(3/2)+a*b*(d*(c^2*x^2+1))^(1/2)*c^4*d/(c^2*x^2+1)*arcsinh(c*x)*x^3-1/4*a*b*(d*(c^2*x^2+1))^(1/2)*c*d/(c^2*x^2
+1)^(1/2)+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c*d+2*b^2*(d*
(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c*d-1/2*b^2*(d*(c^2*x^2+1))^(1/2
)*c^3*d/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^2+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*c^4*d/(c^2*x^2+1)*arcsinh(c*x)^2*x^3-
1/2*b^2*(d*(c^2*x^2+1))^(1/2)*c^2*d/(c^2*x^2+1)*arcsinh(c*x)^2*x-b^2*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*d/x/
(c^2*x^2+1)+1/4*b^2*(d*(c^2*x^2+1))^(1/2)*c^4*d/(c^2*x^2+1)*x^3+1/4*b^2*(d*(c^2*x^2+1))^(1/2)*c^2*d/(c^2*x^2+1
)*x+1/2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^3*c*d-b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(
1/2)*arcsinh(c*x)^2*c*d+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c*d-1/4
*b^2*(d*(c^2*x^2+1))^(1/2)*c*d/(c^2*x^2+1)^(1/2)*arcsinh(c*x)+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*po
lylog(2,-c*x-(c^2*x^2+1)^(1/2))*c*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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