3.51 \(\int \frac {x^6}{\sinh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac {5 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac {x^6 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

[Out]

-5/64*Shi(arcsinh(a*x))/a^7+27/64*Shi(3*arcsinh(a*x))/a^7-25/64*Shi(5*arcsinh(a*x))/a^7+7/64*Shi(7*arcsinh(a*x
))/a^7-x^6*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)

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Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5665, 3298} \[ -\frac {5 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac {x^6 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcSinh[a*x]^2,x]

[Out]

-((x^6*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) - (5*SinhIntegral[ArcSinh[a*x]])/(64*a^7) + (27*SinhIntegral[3*Arc
Sinh[a*x]])/(64*a^7) - (25*SinhIntegral[5*ArcSinh[a*x]])/(64*a^7) + (7*SinhIntegral[7*ArcSinh[a*x]])/(64*a^7)

Rule 3298

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 5665

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {x^6}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^6 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {5 \sinh (x)}{64 x}+\frac {27 \sinh (3 x)}{64 x}-\frac {25 \sinh (5 x)}{64 x}+\frac {7 \sinh (7 x)}{64 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^7}\\ &=-\frac {x^6 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {5 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}+\frac {7 \operatorname {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}-\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}+\frac {27 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac {x^6 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {5 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 85, normalized size = 1.04 \[ -\frac {64 a^6 x^6 \sqrt {a^2 x^2+1}+5 \sinh ^{-1}(a x) \text {Shi}\left (\sinh ^{-1}(a x)\right )-27 \sinh ^{-1}(a x) \text {Shi}\left (3 \sinh ^{-1}(a x)\right )+25 \sinh ^{-1}(a x) \text {Shi}\left (5 \sinh ^{-1}(a x)\right )-7 \sinh ^{-1}(a x) \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7 \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/ArcSinh[a*x]^2,x]

[Out]

-1/64*(64*a^6*x^6*Sqrt[1 + a^2*x^2] + 5*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] - 27*ArcSinh[a*x]*SinhIntegral
[3*ArcSinh[a*x]] + 25*ArcSinh[a*x]*SinhIntegral[5*ArcSinh[a*x]] - 7*ArcSinh[a*x]*SinhIntegral[7*ArcSinh[a*x]])
/(a^7*ArcSinh[a*x])

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fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsinh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^6/arcsinh(a*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsinh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^6/arcsinh(a*x)^2, x)

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maple [A]  time = 0.21, size = 104, normalized size = 1.27 \[ \frac {\frac {5 \sqrt {a^{2} x^{2}+1}}{64 \arcsinh \left (a x \right )}-\frac {5 \Shi \left (\arcsinh \left (a x \right )\right )}{64}-\frac {9 \cosh \left (3 \arcsinh \left (a x \right )\right )}{64 \arcsinh \left (a x \right )}+\frac {27 \Shi \left (3 \arcsinh \left (a x \right )\right )}{64}+\frac {5 \cosh \left (5 \arcsinh \left (a x \right )\right )}{64 \arcsinh \left (a x \right )}-\frac {25 \Shi \left (5 \arcsinh \left (a x \right )\right )}{64}-\frac {\cosh \left (7 \arcsinh \left (a x \right )\right )}{64 \arcsinh \left (a x \right )}+\frac {7 \Shi \left (7 \arcsinh \left (a x \right )\right )}{64}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/arcsinh(a*x)^2,x)

[Out]

1/a^7*(5/64/arcsinh(a*x)*(a^2*x^2+1)^(1/2)-5/64*Shi(arcsinh(a*x))-9/64/arcsinh(a*x)*cosh(3*arcsinh(a*x))+27/64
*Shi(3*arcsinh(a*x))+5/64/arcsinh(a*x)*cosh(5*arcsinh(a*x))-25/64*Shi(5*arcsinh(a*x))-1/64/arcsinh(a*x)*cosh(7
*arcsinh(a*x))+7/64*Shi(7*arcsinh(a*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{3} x^{9} + a x^{7} + {\left (a^{2} x^{8} + x^{6}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {7 \, a^{5} x^{10} + 14 \, a^{3} x^{8} + 7 \, a x^{6} + {\left (7 \, a^{3} x^{8} + 5 \, a x^{6}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (14 \, a^{4} x^{9} + 19 \, a^{2} x^{7} + 6 \, x^{5}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} + a^{2} x\right )} \sqrt {a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^9 + a*x^7 + (a^2*x^8 + x^6)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt
(a^2*x^2 + 1))) + integrate((7*a^5*x^10 + 14*a^3*x^8 + 7*a*x^6 + (7*a^3*x^8 + 5*a*x^6)*(a^2*x^2 + 1) + (14*a^4
*x^9 + 19*a^2*x^7 + 6*x^5)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3*x^2 + 2*a^3*x^2 + 2*(a^4*x^3 + a^2
*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqrt(a^2*x^2 + 1))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^6}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/asinh(a*x)^2,x)

[Out]

int(x^6/asinh(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/asinh(a*x)**2,x)

[Out]

Integral(x**6/asinh(a*x)**2, x)

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