3.178 \(\int \frac {x^3 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\)

Optimal. Leaf size=70 \[ \frac {2 x}{3 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}-\frac {x^3}{9 a} \]

[Out]

2/3*x/a^3-1/9*x^3/a-2/3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^4+1/3*x^2*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^2

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Rubi [A]  time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5758, 5717, 8, 30} \[ \frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}+\frac {2 x}{3 a^3}-\frac {x^3}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(2*x)/(3*a^3) - x^3/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/
(3*a^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5717

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

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(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[m, 1] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^3 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}-\frac {2 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {\int x^2 \, dx}{3 a}\\ &=-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}+\frac {2 \int 1 \, dx}{3 a^3}\\ &=\frac {2 x}{3 a^3}-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{3 a^2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 48, normalized size = 0.69 \[ \frac {-a^3 x^3+3 \left (a^2 x^2-2\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)+6 a x}{9 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(6*a*x - a^3*x^3 + 3*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4)

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fricas [A]  time = 0.71, size = 55, normalized size = 0.79 \[ -\frac {a^{3} x^{3} - 3 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 6 \, a x}{9 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/9*(a^3*x^3 - 3*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1)) - 6*a*x)/a^4

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.00, size = 82, normalized size = 1.17 \[ \frac {3 \arcsinh \left (a x \right ) x^{4} a^{4}-3 \arcsinh \left (a x \right ) x^{2} a^{2}-\sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}-6 \arcsinh \left (a x \right )+6 \sqrt {a^{2} x^{2}+1}\, x a}{9 a^{4} \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x)

[Out]

1/9/a^4*(3*arcsinh(a*x)*x^4*a^4-3*arcsinh(a*x)*x^2*a^2-(a^2*x^2+1)^(1/2)*x^3*a^3-6*arcsinh(a*x)+6*(a^2*x^2+1)^
(1/2)*x*a)/(a^2*x^2+1)^(1/2)

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maxima [A]  time = 0.40, size = 59, normalized size = 0.84 \[ -\frac {1}{9} \, a {\left (\frac {x^{3}}{a^{2}} - \frac {6 \, x}{a^{4}}\right )} + \frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/9*a*(x^3/a^2 - 6*x/a^4) + 1/3*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*asinh(a*x))/(a^2*x^2 + 1)^(1/2),x)

[Out]

int((x^3*asinh(a*x))/(a^2*x^2 + 1)^(1/2), x)

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sympy [A]  time = 1.20, size = 65, normalized size = 0.93 \[ \begin {cases} - \frac {x^{3}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{2}} + \frac {2 x}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**3/(9*a) + x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)/(3*a**2) + 2*x/(3*a**3) - 2*sqrt(a**2*x**2 + 1)*a
sinh(a*x)/(3*a**4), Ne(a, 0)), (0, True))

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