3.3.73 \(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))^2}{x^4} \, dx\) [273]

Optimal. Leaf size=378 \[ -\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {4 b^2 c^3 d \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \]

[Out]

-1/3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x^3-1/3*b^2*c^2*d*(c^2*d*x^2+d)^(1/2)/x-c^2*d*(a+b*arcsinh(c*x))
^2*(c^2*d*x^2+d)^(1/2)/x+1/3*b^2*c^3*d*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+4/3*c^3*d*(a+b*arcsi
nh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/3*c^3*d*(a+b*arcsinh(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/(c^2*x^2
+1)^(1/2)+8/3*b*c^3*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)-4/3*b^2*c^3*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)-1/3*b*c*d*(a+b*a
rcsinh(c*x))*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/x^2

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Rubi [A]
time = 0.45, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5807, 5805, 5775, 3797, 2221, 2317, 2438, 5783, 5802, 283, 221} \begin {gather*} -\frac {c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {c^2 x^2+1}}+\frac {4 c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {8 b c^3 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 )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d}}{3 x}-\frac {4 b^2 c^3 d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {b^2 c^3 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \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^4,x]

[Out]

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

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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 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 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 5802

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((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])/(f*(m + 1))), x] + (-Dist[b*c*(d^p/(f*(m + 1))), Int[(f*x)^(m + 1
)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*A
rcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

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]

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^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (c^2 d\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^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^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 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 (c^4 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}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 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^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \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}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}-\frac {\left (4 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (4 b c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \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}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \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}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \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}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}+\frac {4 b^2 c^3 d \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.95, size = 458, normalized size = 1.21 \begin {gather*} \frac {-a b c d x \sqrt {d+c^2 d x^2}-a^2 d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-4 a^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+b d \sqrt {d+c^2 d x^2} \left (3 a c^3 x^3-b \left (-4 c^3 x^3+\sqrt {1+c^2 x^2}+4 c^2 x^2 \sqrt {1+c^2 x^2}\right )\right ) \sinh ^{-1}(c x)^2+b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^3+b d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (-b c x-2 a \sqrt {1+c^2 x^2} \left (1+4 c^2 x^2\right )+8 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+8 a b c^3 d x^3 \sqrt {d+c^2 d x^2} \log (c x)+3 a^2 c^3 d^{3/2} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-4 b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 x^3 \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^4,x]

[Out]

(-(a*b*c*d*x*Sqrt[d + c^2*d*x^2]) - a^2*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 4*a^2*c^2*d*x^2*Sqrt[1 + c^2
*x^2]*Sqrt[d + c^2*d*x^2] - b^2*c^2*d*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + b*d*Sqrt[d + c^2*d*x^2]*(3*a
*c^3*x^3 - b*(-4*c^3*x^3 + Sqrt[1 + c^2*x^2] + 4*c^2*x^2*Sqrt[1 + c^2*x^2]))*ArcSinh[c*x]^2 + b^2*c^3*d*x^3*Sq
rt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(-(b*c*x) - 2*a*Sqrt[1 + c^2*x^2]*(1 +
4*c^2*x^2) + 8*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])]) + 8*a*b*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 3*a^2
*c^3*d^(3/2)*x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 4*b^2*c^3*d*x^3*Sqrt[d + c^2*d*x
^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*x^3*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. \(2795\) vs. \(2(350)=700\).
time = 4.06, size = 2796, normalized size = 7.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2796\)

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^4,x,method=_RETURNVERBOSE)

[Out]

1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^3*c^3*d+8/3*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^3*d-1/3*a^2/d/x^3*(c^2*d*x^2+d)^(5/2)+2/3*a^2*c^4*x*(c^2*d*x^2+d)^(
3/2)+a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3*d-16/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+
1)^(1/2)*arcsinh(c*x)*c^3*d+16/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3*c^6+4/3*a*b*(d*(c^2*
x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x*c^4-3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2*x^2
+1)^(1/2)*c^3+8/3*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^3*d-64*a*b*(d*
(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8+64*a*b*(d*(c^2*x^2+1))^(1/2)*d/
(24*c^4*x^4+9*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^7-104*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*
c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6+24*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x
^2+1)^(1/2)*arcsinh(c*x)*c^5-146/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(
c*x)*c^4-28/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)*c^2-16/3*a*b*(d*
(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-20/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4
+9*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-8*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)
*c^5+8/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-4/3*a*b*(d*(c
^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^
2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-32*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsi
nh(c*x)^2*c^8-52*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)^2*c^6+12*b^
2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^5-8*b^2*(d*(c^2*x^2+
1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(2
4*c^4*x^4+9*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c+32*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x
^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^7-73/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x/(c^
2*x^2+1)*arcsinh(c*x)^2*c^4-2/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c
*x)-2/3*a^2*c^2/d/x*(c^2*d*x^2+d)^(5/2)+a^2*c^4*d*x*(c^2*d*x^2+d)^(1/2)+a^2*c^4*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)-4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x/(c^2*x^2+1)*arcsi
nh(c*x)*c^4-14/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)^2*c^2-16/3*b^
2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-20/3*b^2*(d*(c^2*x^2+1))^(
1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6+4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9
*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3-3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2*x
^2+1)^(1/2)*arcsinh(c*x)*c^3+8/3*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^3*d-20/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-29/3*b^2*(d*(c^2*x
^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6+4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*
x^2+1)*x*arcsinh(c*x)*c^4+16/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3*arcsinh(c*x)*c^6+8*b^2
*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7+3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*
c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+8/3*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^3*d-10/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/
3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x/(c^2*x^2+1)*c^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c
^4*x^4+9*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)^2-4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^
3*c^6+1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3-8/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3*d+8/3*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^3*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^4,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^4,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^4, 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^{4}}\, 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**4,x)

[Out]

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

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(3/2))/x^4, x)

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