3.4.38 \(\int \frac {\sinh ^{-1}(a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [338]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 218, normalized size of antiderivative = 1.00, 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 = {5787, 5797, 3799, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x) \text {Li}_2\left (-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 {Li}_3\left (-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}} \end {gather*}

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 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 5787

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/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[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 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 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 {\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.18, size = 133, normalized size = 0.61 \begin {gather*} \frac {2 a x \sinh ^{-1}(a x)^3-2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \left (\sinh ^{-1}(a x)+3 \log \left (1+e^{-2 \sinh ^{-1}(a x)}\right )\right )+6 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )}{2 a c \sqrt {c \left (1+a^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)])

________________________________________________________________________________________

Maple [A]
time = 2.58, size = 262, normalized size = 1.20

method	result	size
default	\(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x -\sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right )^{3}}{a \,c^{2} \left (a^{2} x^{2}+1\right )}+\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{3}}{\sqrt {a^{2} x^{2}+1}\, a \,c^{2}}-\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{2}}-\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{2}}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \polylog \left (3, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{2 \sqrt {a^{2} x^{2}+1}\, a \,c^{2}}\)	\(262\)

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,method=_RETURNVERBOSE)

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

________________________________________________________________________________________

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(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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [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(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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________