3.338 \(\int \frac{\sinh ^{-1}(a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a c \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.188048, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5687, 5714, 3718, 2190, 2531, 2282, 6589} \[ -\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a c \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5687

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSinh
[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

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]

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a c \sqrt{c+a^2 c x^2}}-\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}+\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{2 a c \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{2 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.260056, size = 133, normalized size = 0.61 \[ \frac{6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )-2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \left (\sinh ^{-1}(a x)+3 \log \left (e^{-2 \sinh ^{-1}(a x)}+1\right )\right )+2 a x \sinh ^{-1}(a x)^3}{2 a c \sqrt{c \left (a^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a*x*ArcSinh[a*x]^3 - 2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*(ArcSinh[a*x] + 3*Log[1 + E^(-2*ArcSinh[a*x])]) + 6
*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(-2*ArcSi
nh[a*x])])/(2*a*c*Sqrt[c*(1 + a^2*x^2)])

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Maple [A]  time = 0.121, size = 262, normalized size = 1.2 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ( ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }+2\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{2}}}-3\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+ \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{2}}}-3\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,- \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{2}}}+{\frac{3}{2\,a{c}^{2}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 3,- \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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