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

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Rubi [A]  time = 0.16, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.08, 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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fricas [B]  time = 0.70, size = 96, normalized size = 1.88 \[ \frac {b^{2} c x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + {\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt {c^{2} x^{2} - 1} a b + 2 \, {\left (a b c x - \sqrt {c^{2} x^{2} - 1} b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{c} \]

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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giac [B]  time = 0.48, size = 111, normalized size = 2.18 \[ 2 \, {\left (x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{c}\right )} a b + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, c {\left (\frac {x}{c} - \frac {\sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \]

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

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maple [A]  time = 0.07, size = 78, normalized size = 1.53 \[ \frac {a^{2} c x +b^{2} \left (\mathrm {arccosh}\left (c x \right )^{2} c x -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+2 a b \left (c x \,\mathrm {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2*c*x+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 = 0.31, size = 72, normalized size = 1.41 \[ b^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, b^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b}{c} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^2,x)

[Out]

int((a + b*acosh(c*x))^2, x)

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sympy [A]  time = 0.27, size = 88, normalized size = 1.73 \[ \begin {cases} a^{2} x + 2 a b x \operatorname {acosh}{\left (c x \right )} - \frac {2 a b \sqrt {c^{2} x^{2} - 1}}{c} + b^{2} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {i \pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \]

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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