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

Optimal. Leaf size=51 \[ -\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2+2 b^2 x \]

[Out]

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

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Rubi [A]  time = 0.156815, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5654, 5718, 8} \[ -\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2+2 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^2,x]

[Out]

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0870446, size = 84, normalized size = 1.65 \[ x \left (a^2+2 b^2\right )-\frac{2 a b \sqrt{c x-1} \sqrt{c x+1}}{c}+\frac{2 b \cosh ^{-1}(c x) \left (a c x-b \sqrt{c x-1} \sqrt{c x+1}\right )}{c}+b^2 x \cosh ^{-1}(c x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])^2,x]

[Out]

(a^2 + 2*b^2)*x - (2*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + (2*b*(a*c*x - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])*ArcCo
sh[c*x])/c + b^2*x*ArcCosh[c*x]^2

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Maple [A]  time = 0.04, size = 78, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ( cx{a}^{2}+{b}^{2} \left ( \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cx-2\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+2\,cx \right ) +2\,ab \left ( cx{\rm arccosh} \left (cx\right )-\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17358, size = 97, normalized size = 1.9 \begin{align*} b^{2} x \operatorname{arcosh}\left (c x\right )^{2} + 2 \, b^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} - 1} \operatorname{arcosh}\left (c x\right )}{c}\right )} + a^{2} x + \frac{2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} a b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*x*arccosh(c*x)^2 + 2*b^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x
^2 - 1))*a*b/c

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Fricas [B]  time = 1.71647, size = 212, normalized size = 4.16 \begin{align*} \frac{b^{2} c x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} +{\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt{c^{2} x^{2} - 1} a b + 2 \,{\left (a b c x - \sqrt{c^{2} x^{2} - 1} b^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.363591, size = 88, normalized size = 1.73 \begin{align*} \begin{cases} a^{2} x + 2 a b x \operatorname{acosh}{\left (c x \right )} - \frac{2 a b \sqrt{c^{2} x^{2} - 1}}{c} + b^{2} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac{2 b^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} & \text{for}\: c \neq 0 \\x \left (a + \frac{i \pi b}{2}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.32592, size = 150, normalized size = 2.94 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} a b +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} - 1} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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