3.4.81 \(\int \frac {x^3}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx\) [381]

Optimal. Leaf size=27 \[ -\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5819, 3393, 3379} \begin {gather*} \frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*SinhIntegral[ArcSinh[a*x]])/(4*a^4) + SinhIntegral[3*ArcSinh[a*x]]/(4*a^4)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5819

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {i \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 22, normalized size = 0.81 \begin {gather*} \frac {-3 \text {Shi}\left (\sinh ^{-1}(a x)\right )+\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*SinhIntegral[ArcSinh[a*x]] + SinhIntegral[3*ArcSinh[a*x]])/(4*a^4)

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Maple [A]
time = 2.79, size = 23, normalized size = 0.85

method	result	size
default	\(-\frac {3 \hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )-\hyperbolicSineIntegral \left (3 \arcsinh \left (a x \right )\right )}{4 a^{4}}\)	\(23\)

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,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

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

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

[Out]

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

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

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]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^3}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \end {gather*}

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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