3.342 \(\int \frac{x^4 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{45 x^2}{128 a^3}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{8 a^4}-\frac{45 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{64 a^4}+\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}+\frac{45 \sinh ^{-1}(a x)^2}{128 a^5}-\frac{3 x^4}{128 a}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a} \]

[Out]

(45*x^2)/(128*a^3) - (3*x^4)/(128*a) - (45*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(64*a^4) + (3*x^3*Sqrt[1 + a^2*x^
2]*ArcSinh[a*x])/(32*a^2) + (45*ArcSinh[a*x]^2)/(128*a^5) + (9*x^2*ArcSinh[a*x]^2)/(16*a^3) - (3*x^4*ArcSinh[a
*x]^2)/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a^2
) + (3*ArcSinh[a*x]^4)/(32*a^5)

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Rubi [A]  time = 0.498473, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5758, 5675, 5661, 30} \[ \frac{45 x^2}{128 a^3}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{8 a^4}-\frac{45 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{64 a^4}+\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}+\frac{45 \sinh ^{-1}(a x)^2}{128 a^5}-\frac{3 x^4}{128 a}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*x^2)/(128*a^3) - (3*x^4)/(128*a) - (45*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(64*a^4) + (3*x^3*Sqrt[1 + a^2*x^
2]*ArcSinh[a*x])/(32*a^2) + (45*ArcSinh[a*x]^2)/(128*a^5) + (9*x^2*ArcSinh[a*x]^2)/(16*a^3) - (3*x^4*ArcSinh[a
*x]^2)/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a^2
) + (3*ArcSinh[a*x]^4)/(32*a^5)

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]

Rule 5675

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

Rule 5661

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 30

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

Rubi steps

\begin{align*} \int \frac{x^4 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{4 a^2}-\frac{3 \int \frac{x^2 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2}-\frac{3 \int x^3 \sinh ^{-1}(a x)^2 \, dx}{4 a}\\ &=-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3}{8} \int \frac{x^4 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx+\frac{3 \int \frac{\sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{8 a^4}+\frac{9 \int x \sinh ^{-1}(a x)^2 \, dx}{8 a^3}\\ &=\frac{3 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}-\frac{9 \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{32 a^2}-\frac{9 \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2}-\frac{3 \int x^3 \, dx}{32 a}\\ &=-\frac{3 x^4}{128 a}-\frac{45 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{64 a^4}+\frac{3 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}+\frac{9 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{64 a^4}+\frac{9 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{16 a^4}+\frac{9 \int x \, dx}{64 a^3}+\frac{9 \int x \, dx}{16 a^3}\\ &=\frac{45 x^2}{128 a^3}-\frac{3 x^4}{128 a}-\frac{45 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{64 a^4}+\frac{3 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{32 a^2}+\frac{45 \sinh ^{-1}(a x)^2}{128 a^5}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0768526, size = 121, normalized size = 0.65 \[ \frac{-3 a^4 x^4+45 a^2 x^2+16 a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)^3+6 a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-15\right ) \sinh ^{-1}(a x)+\left (-24 a^4 x^4+72 a^2 x^2+45\right ) \sinh ^{-1}(a x)^2+12 \sinh ^{-1}(a x)^4}{128 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*a^2*x^2 - 3*a^4*x^4 + 6*a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2)*ArcSinh[a*x] + (45 + 72*a^2*x^2 - 24*a^4*x
^4)*ArcSinh[a*x]^2 + 16*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^3 + 12*ArcSinh[a*x]^4)/(128*a^5)

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Maple [A]  time = 0.061, size = 156, normalized size = 0.8 \begin{align*}{\frac{1}{128\,{a}^{5}} \left ( 32\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\sqrt{{a}^{2}{x}^{2}+1}{a}^{3}{x}^{3}-24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}+12\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}{a}^{3}{x}^{3}-3\,{x}^{4}{a}^{4}-48\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax\sqrt{{a}^{2}{x}^{2}+1}+72\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}+12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}-90\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax+45\,{a}^{2}{x}^{2}+45\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+48 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)

[Out]

1/128*(32*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)*a^3*x^3-24*arcsinh(a*x)^2*x^4*a^4+12*arcsinh(a*x)*(a^2*x^2+1)^(1/2)
*a^3*x^3-3*x^4*a^4-48*arcsinh(a*x)^3*a*x*(a^2*x^2+1)^(1/2)+72*arcsinh(a*x)^2*a^2*x^2+12*arcsinh(a*x)^4-90*arcs
inh(a*x)*(a^2*x^2+1)^(1/2)*a*x+45*a^2*x^2+45*arcsinh(a*x)^2+48)/a^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)

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Fricas [A]  time = 2.16705, size = 383, normalized size = 2.05 \begin{align*} -\frac{3 \, a^{4} x^{4} - 16 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 45 \, a^{2} x^{2} - 12 \, \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} + 3 \,{\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 6 \,{\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{128 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/128*(3*a^4*x^4 - 16*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^3 - 45*a^2*x^2 - 12*
log(a*x + sqrt(a^2*x^2 + 1))^4 + 3*(8*a^4*x^4 - 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(2*a^3*x^3
- 15*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^5

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Sympy [A]  time = 9.19402, size = 185, normalized size = 0.99 \begin{align*} \begin{cases} - \frac{3 x^{4} \operatorname{asinh}^{2}{\left (a x \right )}}{16 a} - \frac{3 x^{4}}{128 a} + \frac{x^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac{3 x^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{32 a^{2}} + \frac{9 x^{2} \operatorname{asinh}^{2}{\left (a x \right )}}{16 a^{3}} + \frac{45 x^{2}}{128 a^{3}} - \frac{3 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{8 a^{4}} - \frac{45 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{64 a^{4}} + \frac{3 \operatorname{asinh}^{4}{\left (a x \right )}}{32 a^{5}} + \frac{45 \operatorname{asinh}^{2}{\left (a x \right )}}{128 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((-3*x**4*asinh(a*x)**2/(16*a) - 3*x**4/(128*a) + x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(4*a**2) + 3
*x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)/(32*a**2) + 9*x**2*asinh(a*x)**2/(16*a**3) + 45*x**2/(128*a**3) - 3*x*sqr
t(a**2*x**2 + 1)*asinh(a*x)**3/(8*a**4) - 45*x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(64*a**4) + 3*asinh(a*x)**4/(32*
a**5) + 45*asinh(a*x)**2/(128*a**5), Ne(a, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)