3.34 \(\int x^3 \sinh ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=194 \[ -\frac {3 \sinh ^{-1}(a x)^4}{32 a^4}-\frac {45 \sinh ^{-1}(a x)^2}{128 a^4}-\frac {45 x^2}{128 a^2}-\frac {9 x^2 \sinh ^{-1}(a x)^2}{16 a^2}-\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{4 a}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{32 a}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{8 a^3}+\frac {45 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{64 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^4+\frac {3}{16} x^4 \sinh ^{-1}(a x)^2+\frac {3 x^4}{128} \]

[Out]

-45/128*x^2/a^2+3/128*x^4-45/128*arcsinh(a*x)^2/a^4-9/16*x^2*arcsinh(a*x)^2/a^2+3/16*x^4*arcsinh(a*x)^2-3/32*a
rcsinh(a*x)^4/a^4+1/4*x^4*arcsinh(a*x)^4+45/64*x*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^3-3/32*x^3*arcsinh(a*x)*(a^2
*x^2+1)^(1/2)/a+3/8*x*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a^3-1/4*x^3*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.50, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5661, 5758, 5675, 30} \[ -\frac {45 x^2}{128 a^2}-\frac {x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{4 a}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{32 a}-\frac {9 x^2 \sinh ^{-1}(a x)^2}{16 a^2}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{8 a^3}+\frac {45 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{64 a^3}-\frac {3 \sinh ^{-1}(a x)^4}{32 a^4}-\frac {45 \sinh ^{-1}(a x)^2}{128 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^4+\frac {3}{16} x^4 \sinh ^{-1}(a x)^2+\frac {3 x^4}{128} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSinh[a*x]^4,x]

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 133, normalized size = 0.69 \[ \frac {4 \left (8 a^4 x^4-3\right ) \sinh ^{-1}(a x)^4+3 a^2 x^2 \left (a^2 x^2-15\right )-16 a x \sqrt {a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)^3-6 a x \sqrt {a^2 x^2+1} \left (2 a^2 x^2-15\right ) \sinh ^{-1}(a x)+3 \left (8 a^4 x^4-24 a^2 x^2-15\right ) \sinh ^{-1}(a x)^2}{128 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcSinh[a*x]^4,x]

[Out]

(3*a^2*x^2*(-15 + a^2*x^2) - 6*a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2)*ArcSinh[a*x] + 3*(-15 - 24*a^2*x^2 + 8*
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 + 4*(-3 + 8*a^4*x^4)*ArcSin
h[a*x]^4)/(128*a^4)

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fricas [A]  time = 0.44, size = 176, normalized size = 0.91 \[ \frac {3 \, a^{4} x^{4} + 4 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 16 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 45 \, a^{2} x^{2} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{128 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

1/128*(3*a^4*x^4 + 4*(8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 + 1))^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 + 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^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)^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 = 0.38, size = 172, normalized size = 0.89 \[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{4}}{4}-\frac {a^{3} x^{3} \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{4}+\frac {3 a x \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \arcsinh \left (a x \right )^{4}}{32}+\frac {3 a^{4} x^{4} \arcsinh \left (a x \right )^{2}}{16}-\frac {3 a^{3} x^{3} \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{32}+\frac {45 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{64}+\frac {27 \arcsinh \left (a x \right )^{2}}{128}+\frac {3 a^{4} x^{4}}{128}-\frac {45 a^{2} x^{2}}{128}-\frac {45}{128}-\frac {9 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{2}}{16}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsinh(a*x)^4,x)

[Out]

1/a^4*(1/4*a^4*x^4*arcsinh(a*x)^4-1/4*a^3*x^3*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)+3/8*a*x*arcsinh(a*x)^3*(a^2*x^2
+1)^(1/2)-3/32*arcsinh(a*x)^4+3/16*a^4*x^4*arcsinh(a*x)^2-3/32*a^3*x^3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)+45/64*ar
csinh(a*x)*(a^2*x^2+1)^(1/2)*a*x+27/128*arcsinh(a*x)^2+3/128*a^4*x^4-45/128*a^2*x^2-45/128-9/16*(a^2*x^2+1)*ar
csinh(a*x)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - \int \frac {{\left (a^{3} x^{6} + \sqrt {a^{2} x^{2} + 1} a^{2} x^{5} + a x^{4}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{3} x^{3} + a x + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/4*x^4*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate((a^3*x^6 + sqrt(a^2*x^2 + 1)*a^2*x^5 + a*x^4)*log(a*x + sqr
t(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 \[ \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*asinh(a*x)^4,x)

[Out]

int(x^3*asinh(a*x)^4, x)

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sympy [A]  time = 5.56, size = 190, normalized size = 0.98 \[ \begin {cases} \frac {x^{4} \operatorname {asinh}^{4}{\left (a x \right )}}{4} + \frac {3 x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{16} + \frac {3 x^{4}}{128} - \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{4 a} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{32 a} - \frac {9 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{2}} - \frac {45 x^{2}}{128 a^{2}} + \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{8 a^{3}} + \frac {45 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{64 a^{3}} - \frac {3 \operatorname {asinh}^{4}{\left (a x \right )}}{32 a^{4}} - \frac {45 \operatorname {asinh}^{2}{\left (a x \right )}}{128 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asinh(a*x)**4,x)

[Out]

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

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