3.36 \(\int x \sinh ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=110 \[ -\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{2 a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2+\frac {3 x^2}{4} \]

[Out]

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

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Rubi [A]  time = 0.24, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5661, 5758, 5675, 30} \[ -\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{2 a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2+\frac {3 x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSinh[a*x]^4,x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 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 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 x \sinh ^{-1}(a x)^4 \, dx &=\frac {1}{2} x^2 \sinh ^{-1}(a x)^4-(2 a) \int \frac {x^2 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+3 \int x \sinh ^{-1}(a x)^2 \, dx+\frac {\int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{a}\\ &=\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4-(3 a) \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4+\frac {3 \int x \, dx}{2}+\frac {3 \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a}\\ &=\frac {3 x^2}{4}-\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac {3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac {3}{2} x^2 \sinh ^{-1}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac {\sinh ^{-1}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)^4\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 94, normalized size = 0.85 \[ \frac {3 a^2 x^2+\left (2 a^2 x^2+1\right ) \sinh ^{-1}(a x)^4-4 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3+\left (6 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2-6 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSinh[a*x]^4,x]

[Out]

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

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fricas [A]  time = 0.41, size = 138, normalized size = 1.25 \[ -\frac {4 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 3 \, a^{2} x^{2} + 6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

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

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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*arcsinh(a*x)^4,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 = 0.10, size = 105, normalized size = 0.95 \[ \frac {\frac {\left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{4}}{2}-a x \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}-\frac {\arcsinh \left (a x \right )^{4}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{2}}{2}-\frac {3 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{2}-\frac {3 \arcsinh \left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}+\frac {3}{4}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsinh(a*x)^4,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - \int \frac {2 \, {\left (a^{3} x^{4} + \sqrt {a^{2} x^{2} + 1} a^{2} x^{3} + a x^{2}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{3} x^{3} + a x + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/2*x^2*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate(2*(a^3*x^4 + sqrt(a^2*x^2 + 1)*a^2*x^3 + a*x^2)*log(a*x + s
qrt(a^2*x^2 + 1))^3/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asinh(a*x)^4,x)

[Out]

int(x*asinh(a*x)^4, x)

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sympy [A]  time = 1.78, size = 104, normalized size = 0.95 \[ \begin {cases} \frac {x^{2} \operatorname {asinh}^{4}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{2} + \frac {3 x^{2}}{4} - \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asinh(a*x)**4,x)

[Out]

Piecewise((x**2*asinh(a*x)**4/2 + 3*x**2*asinh(a*x)**2/2 + 3*x**2/4 - x*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/a -
3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(2*a) + asinh(a*x)**4/(4*a**2) + 3*asinh(a*x)**2/(4*a**2), Ne(a, 0)), (0, T
rue))

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