3.42 \(\int \frac {x^6}{\cosh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=55 \[ \frac {5 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {5 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {\text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a^7} \]

[Out]

5/64*Shi(arccosh(a*x))/a^7+9/64*Shi(3*arccosh(a*x))/a^7+5/64*Shi(5*arccosh(a*x))/a^7+1/64*Shi(7*arccosh(a*x))/
a^7

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Rubi [A]  time = 0.10, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5670, 5448, 3298} \[ \frac {5 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {5 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {\text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcCosh[a*x],x]

[Out]

(5*SinhIntegral[ArcCosh[a*x]])/(64*a^7) + (9*SinhIntegral[3*ArcCosh[a*x]])/(64*a^7) + (5*SinhIntegral[5*ArcCos
h[a*x]])/(64*a^7) + SinhIntegral[7*ArcCosh[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 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^6}{\cosh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^6(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^7}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 x}+\frac {9 \sinh (3 x)}{64 x}+\frac {5 \sinh (5 x)}{64 x}+\frac {\sinh (7 x)}{64 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^7}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {5 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {5 \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^7}\\ &=\frac {5 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a^7}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {5 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a^7}+\frac {\text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a^7}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 40, normalized size = 0.73 \[ \frac {5 \text {Shi}\left (\cosh ^{-1}(a x)\right )+9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+5 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )+\text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/ArcCosh[a*x],x]

[Out]

(5*SinhIntegral[ArcCosh[a*x]] + 9*SinhIntegral[3*ArcCosh[a*x]] + 5*SinhIntegral[5*ArcCosh[a*x]] + SinhIntegral
[7*ArcCosh[a*x]])/(64*a^7)

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^6/arccosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6/arccosh(a*x), x)

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maple [A]  time = 0.11, size = 40, normalized size = 0.73 \[ \frac {\frac {5 \Shi \left (\mathrm {arccosh}\left (a x \right )\right )}{64}+\frac {9 \Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{64}+\frac {5 \Shi \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{64}+\frac {\Shi \left (7 \,\mathrm {arccosh}\left (a x \right )\right )}{64}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/arccosh(a*x),x)

[Out]

1/a^7*(5/64*Shi(arccosh(a*x))+9/64*Shi(3*arccosh(a*x))+5/64*Shi(5*arccosh(a*x))+1/64*Shi(7*arccosh(a*x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6/arccosh(a*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^6}{\mathrm {acosh}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/acosh(a*x),x)

[Out]

int(x^6/acosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/acosh(a*x),x)

[Out]

Integral(x**6/acosh(a*x), x)

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