3.15 \(\int x \cosh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=64 \[ -\frac {\cosh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^2-\frac {x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{2 a}+\frac {x^2}{4} \]

[Out]

1/4*x^2-1/4*arccosh(a*x)^2/a^2+1/2*x^2*arccosh(a*x)^2-1/2*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]  time = 0.25, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5662, 5759, 5676, 30} \[ -\frac {\cosh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^2-\frac {x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{2 a}+\frac {x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCosh[a*x]^2,x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rubi steps

\begin {align*} \int x \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{2} x^2 \cosh ^{-1}(a x)^2-a \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^2+\frac {\int x \, dx}{2}-\frac {\int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 a}\\ &=\frac {x^2}{4}-\frac {x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a}-\frac {\cosh ^{-1}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 58, normalized size = 0.91 \[ \frac {a^2 x^2+\left (2 a^2 x^2-1\right ) \cosh ^{-1}(a x)^2-2 a x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCosh[a*x]^2,x]

[Out]

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

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fricas [A]  time = 0.59, size = 73, normalized size = 1.14 \[ \frac {a^{2} x^{2} - 2 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccosh(a*x)^2,x, algorithm="fricas")

[Out]

1/4*(a^2*x^2 - 2*sqrt(a^2*x^2 - 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1)) + (2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 -
1))^2)/a^2

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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*arccosh(a*x)^2,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 = 0.03, size = 58, normalized size = 0.91 \[ \frac {\frac {a^{2} x^{2} \mathrm {arccosh}\left (a x \right )^{2}}{2}-\frac {\mathrm {arccosh}\left (a x \right ) a x \sqrt {a x -1}\, \sqrt {a x +1}}{2}-\frac {\mathrm {arccosh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{4}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccosh(a*x)^2,x)

[Out]

1/a^2*(1/2*a^2*x^2*arccosh(a*x)^2-1/2*arccosh(a*x)*a*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/4*arccosh(a*x)^2+1/4*a^2*
x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccosh(a*x)^2,x, algorithm="maxima")

[Out]

1/2*x^2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 - integrate((a^3*x^4 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^3 -
a*x^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*acosh(a*x)^2,x)

[Out]

int(x*acosh(a*x)^2, x)

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sympy [A]  time = 0.49, size = 60, normalized size = 0.94 \[ \begin {cases} \frac {x^{2} \operatorname {acosh}^{2}{\left (a x \right )}}{2} + \frac {x^{2}}{4} - \frac {x \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{2 a} - \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acosh(a*x)**2,x)

[Out]

Piecewise((x**2*acosh(a*x)**2/2 + x**2/4 - x*sqrt(a**2*x**2 - 1)*acosh(a*x)/(2*a) - acosh(a*x)**2/(4*a**2), Ne
(a, 0)), (-pi**2*x**2/8, True))

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