3.44 \(\int (c+d x^2) \cosh ^{-1}(a x) \, dx\)

Optimal. Leaf size=84 \[ -\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)-\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a} \]

[Out]

c*x*arccosh(a*x)+1/3*d*x^3*arccosh(a*x)-1/9*(9*a^2*c+2*d)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/9*d*x^2*(a*x-1)^(1
/2)*(a*x+1)^(1/2)/a

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Rubi [A]  time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5705, 460, 74} \[ -\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \cosh ^{-1}(a x)-\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a}+\frac {1}{3} d x^3 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)*ArcCosh[a*x],x]

[Out]

-((9*a^2*c + 2*d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) - (d*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + c*x*Arc
Cosh[a*x] + (d*x^3*ArcCosh[a*x])/3

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \left (c+d x^2\right ) \cosh ^{-1}(a x) \, dx &=c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)-a \int \frac {x \left (c+\frac {d x^2}{3}\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)+\frac {1}{9} \left (a \left (-9 c-\frac {2 d}{a^2}\right )\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 60, normalized size = 0.71 \[ \cosh ^{-1}(a x) \left (c x+\frac {d x^3}{3}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (a^2 \left (9 c+d x^2\right )+2 d\right )}{9 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)*ArcCosh[a*x],x]

[Out]

-1/9*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2*d + a^2*(9*c + d*x^2)))/a^3 + (c*x + (d*x^3)/3)*ArcCosh[a*x]

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fricas [A]  time = 0.57, size = 71, normalized size = 0.85 \[ \frac {3 \, {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} d x^{2} + 9 \, a^{2} c + 2 \, d\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*(a^3*d*x^3 + 3*a^3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (a^2*d*x^2 + 9*a^2*c + 2*d)*sqrt(a^2*x^2 - 1))/a
^3

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giac [A]  time = 0.31, size = 70, normalized size = 0.83 \[ \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d}{9 \, a^{3}} - \frac {\sqrt {a^{2} x^{2} - 1} {\left (3 \, a^{2} c + d\right )}}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccosh(a*x),x, algorithm="giac")

[Out]

1/3*(d*x^3 + 3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/9*(a^2*x^2 - 1)^(3/2)*d/a^3 - 1/3*sqrt(a^2*x^2 - 1)*(3*a^
2*c + d)/a^3

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maple [A]  time = 0.01, size = 62, normalized size = 0.74 \[ \frac {\frac {a \,\mathrm {arccosh}\left (a x \right ) d \,x^{3}}{3}+\mathrm {arccosh}\left (a x \right ) c a x -\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} d \,x^{2}+9 a^{2} c +2 d \right )}{9 a^{2}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)*arccosh(a*x),x)

[Out]

1/a*(1/3*a*arccosh(a*x)*d*x^3+arccosh(a*x)*c*a*x-1/9/a^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(a^2*d*x^2+9*a^2*c+2*d))

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maxima [A]  time = 0.31, size = 74, normalized size = 0.88 \[ -\frac {1}{9} \, {\left (\frac {\sqrt {a^{2} x^{2} - 1} d x^{2}}{a^{2}} + \frac {9 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1} d}{a^{4}}\right )} a + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcosh}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(sqrt(a^2*x^2 - 1)*d*x^2/a^2 + 9*sqrt(a^2*x^2 - 1)*c/a^2 + 2*sqrt(a^2*x^2 - 1)*d/a^4)*a + 1/3*(d*x^3 + 3*
c*x)*arccosh(a*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)*(c + d*x^2),x)

[Out]

int(acosh(a*x)*(c + d*x^2), x)

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sympy [A]  time = 0.51, size = 90, normalized size = 1.07 \[ \begin {cases} c x \operatorname {acosh}{\left (a x \right )} + \frac {d x^{3} \operatorname {acosh}{\left (a x \right )}}{3} - \frac {c \sqrt {a^{2} x^{2} - 1}}{a} - \frac {d x^{2} \sqrt {a^{2} x^{2} - 1}}{9 a} - \frac {2 d \sqrt {a^{2} x^{2} - 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c x + \frac {d x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)*acosh(a*x),x)

[Out]

Piecewise((c*x*acosh(a*x) + d*x**3*acosh(a*x)/3 - c*sqrt(a**2*x**2 - 1)/a - d*x**2*sqrt(a**2*x**2 - 1)/(9*a) -
2*d*sqrt(a**2*x**2 - 1)/(9*a**3), Ne(a, 0)), (I*pi*(c*x + d*x**3/3)/2, True))

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