∫ x____ cosh−1 (ax)4 dx [68]

3.1.68 \(\int \frac {x}{\cosh ^{-1}(a x)^4} \, dx\) [68]

Optimal. Leaf size=105 \[ -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac {x^2}{3 \cosh ^{-1}(a x)^2}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}+\frac {2 \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2} \]

[Out]

1/6/a^2/arccosh(a*x)^2-1/3*x^2/arccosh(a*x)^2+2/3*Chi(2*arccosh(a*x))/a^2-1/3*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/

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

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Rubi [A]
time = 0.25, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5886, 5951, 5885, 3382, 5893} \begin {gather*} \frac {2 \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2}+\frac {1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac {x^2}{3 \cosh ^{-1}(a x)^2}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^3} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x/ArcCosh[a*x]^4,x]

[Out]

-1/3*(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) + 1/(6*a^2*ArcCosh[a*x]^2) - x^2/(3*ArcCosh[a*x]^2) -

(2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]) + (2*CoshIntegral[2*ArcCosh[a*x]])/(3*a^2)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)

+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5885

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((

a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^

(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free

Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((

a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh

[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c

*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]

:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc

Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n

, -1]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_

.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +

e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]

]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,

c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {x}{\cosh ^{-1}(a x)^4} \, dx &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {\int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+\frac {1}{3} (2 a) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac {x^2}{3 \cosh ^{-1}(a x)^2}+\frac {2}{3} \int \frac {x}{\cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac {x^2}{3 \cosh ^{-1}(a x)^2}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {1}{6 a^2 \cosh ^{-1}(a x)^2}-\frac {x^2}{3 \cosh ^{-1}(a x)^2}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}+\frac {2 \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 131, normalized size = 1.25 \begin {gather*} \frac {\frac {2 a x-2 a^3 x^3-\sqrt {-1+a x} \sqrt {1+a x} \left (-1+2 a^2 x^2\right ) \cosh ^{-1}(a x)+\left (4 a x-4 a^3 x^3\right ) \cosh ^{-1}(a x)^2}{\cosh ^{-1}(a x)^3}+4 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{6 a^2 \sqrt {-1+a x} \sqrt {1+a x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x/ArcCosh[a*x]^4,x]

[Out]

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

[a*x]^2)/ArcCosh[a*x]^3 + 4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*CoshIntegral[2*ArcCosh[a*x]])/(6*a^2*Sqrt[-1

+ a*x]*Sqrt[1 + a*x])

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Maple [A]
time = 2.28, size = 60, normalized size = 0.57


method	result	size

derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3 \,\mathrm {arccosh}\left (a x \right )}+\frac {2 \hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3}}{a^{2}}\)	\(60\)
default	\(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3 \,\mathrm {arccosh}\left (a x \right )}+\frac {2 \hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3}}{a^{2}}\)	\(60\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/arccosh(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^2*(-1/6/arccosh(a*x)^3*sinh(2*arccosh(a*x))-1/6/arccosh(a*x)^2*cosh(2*arccosh(a*x))-1/3/arccosh(a*x)*sinh(

2*arccosh(a*x))+2/3*Chi(2*arccosh(a*x)))

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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/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2

)*(a*x - 1)^(5/2) + 2*(5*a^8*x^8 - 9*a^6*x^6 + 4*a^4*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^9*x^9 - 13*a^7*x^7

+ 11*a^5*x^5 - 3*a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a^2*x^2 + 4*(5*a^10*x^10 - 17*a^8*x^8 + 21*a^6*x

^6 - 11*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (4*a^12*x^12 - 20*a^10*x^10 + 40*a^8*x^8 - 40*a^6*x^6 + 20*

a^4*x^4 + 4*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (20*a^8*x^8 - 36*a^6*x^6 + 16*a^4*x^4 + 3*a^

2*x^2 - 3)*(a*x + 1)^2*(a*x - 1)^2 + (40*a^9*x^9 - 104*a^7*x^7 + 88*a^5*x^5 - 21*a^3*x^3 - 3*a*x)*(a*x + 1)^(3

/2)*(a*x - 1)^(3/2) - 4*a^2*x^2 + (40*a^10*x^10 - 136*a^8*x^8 + 168*a^6*x^6 - 91*a^4*x^4 + 22*a^2*x^2 - 3)*(a*

x + 1)*(a*x - 1) + (20*a^11*x^11 - 84*a^9*x^9 + 136*a^7*x^7 - 107*a^5*x^5 + 42*a^3*x^3 - 7*a*x)*sqrt(a*x + 1)*

sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^11*x^11 - 21*a^9*x^9 + 34*a^7*x^7 - 26*a^5*x^

5 + 9*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + (2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*

a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (10*a^8*x^8 - 18*a^6*x^6 + 9*a^4*x^4 - a^2*x

^2)*(a*x + 1)^2*(a*x - 1)^2 + (20*a^9*x^9 - 52*a^7*x^7 + 47*a^5*x^5 - 17*a^3*x^3 + 2*a*x)*(a*x + 1)^(3/2)*(a*x

- 1)^(3/2) - 2*a^2*x^2 + (20*a^10*x^10 - 68*a^8*x^8 + 87*a^6*x^6 - 51*a^4*x^4 + 13*a^2*x^2 - 1)*(a*x + 1)*(a*

x - 1) + (10*a^11*x^11 - 42*a^9*x^9 + 69*a^7*x^7 - 55*a^5*x^5 + 21*a^3*x^3 - 3*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1

))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^12*x^10 - 5*a^10*x^8 + (a*x + 1)^(5/2)*(a*x - 1)^(5/2)*a^7*x^5

+ 10*a^8*x^6 - 10*a^6*x^4 + 5*a^4*x^2 + 5*(a^8*x^6 - a^6*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 10*(a^9*x^7 - 2*a^7*x^

5 + a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^10*x^8 - 3*a^8*x^6 + 3*a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x

- 1) + 5*(a^11*x^9 - 4*a^9*x^7 + 6*a^7*x^5 - 4*a^5*x^3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + s

qrt(a*x + 1)*sqrt(a*x - 1))^3) + integrate(1/6*(8*a^13*x^13 - 48*a^11*x^11 + 8*(a*x + 1)^3*(a*x - 1)^3*a^7*x^7

+ 120*a^9*x^9 - 160*a^7*x^7 + 120*a^5*x^5 + (48*a^8*x^8 - 48*a^6*x^6 + 4*a^4*x^4 - 12*a^2*x^2 + 15)*(a*x + 1)

^(5/2)*(a*x - 1)^(5/2) - 48*a^3*x^3 + 8*(15*a^9*x^9 - 30*a^7*x^7 + 17*a^5*x^5 - 5*a^3*x^3 + 3*a*x)*(a*x + 1)^2

*(a*x - 1)^2 + 2*(80*a^10*x^10 - 240*a^8*x^8 + 252*a^6*x^6 - 104*a^4*x^4 + 3*a^2*x^2 + 9)*(a*x + 1)^(3/2)*(a*x

- 1)^(3/2) + 8*(15*a^11*x^11 - 60*a^9*x^9 + 92*a^7*x^7 - 63*a^5*x^5 + 15*a^3*x^3 + a*x)*(a*x + 1)*(a*x - 1) +

(48*a^12*x^12 - 240*a^10*x^10 + 484*a^8*x^8 - 484*a^6*x^6 + 243*a^4*x^4 - 58*a^2*x^2 + 7)*sqrt(a*x + 1)*sqrt(

a*x - 1) + 8*a*x)/((a^13*x^12 - 6*a^11*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^7*x^6 + 15*a^9*x^8 - 20*a^7*x^6 + 15*a

^5*x^4 + 6*(a^8*x^7 - a^6*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^9*x^8 - 2*a^7*x^6 + a^5*x^4)*(a*x + 1)^

2*(a*x - 1)^2 - 6*a^3*x^2 + 20*(a^10*x^9 - 3*a^8*x^7 + 3*a^6*x^5 - a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) +

15*(a^11*x^10 - 4*a^9*x^8 + 6*a^7*x^6 - 4*a^5*x^4 + a^3*x^2)*(a*x + 1)*(a*x - 1) + 6*(a^12*x^11 - 5*a^10*x^9 +

10*a^8*x^7 - 10*a^6*x^5 + 5*a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(a*

x - 1))), 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/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x/arccosh(a*x)^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/acosh(a*x)**4,x)

[Out]

Integral(x/acosh(a*x)**4, x)

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Giac [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/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x/arccosh(a*x)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/acosh(a*x)^4,x)

[Out]

int(x/acosh(a*x)^4, x)

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