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

Optimal. Leaf size=33 \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5901, 12, 191} \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCosh[-1 + d*x^2],x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 5901

Int[ArcCosh[u_], x_Symbol] :> Simp[x*ArcCosh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 +
u]), x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 33, normalized size = 1.00 \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCosh[-1 + d*x^2],x]

[Out]

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

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fricas [B]  time = 1.22, size = 63, normalized size = 1.91 \[ \frac {b d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) + a d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccosh(d*x^2-1),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.24, size = 67, normalized size = 2.03 \[ {\left (x \log \left (d x^{2} + \sqrt {{\left (d x^{2} - 1\right )}^{2} - 1} - 1\right ) + \frac {2 \, \sqrt {2} \sqrt {-d} \mathrm {sgn}\relax (x)}{d} - \frac {2 \, \sqrt {d^{2} x^{2} - 2 \, d}}{d \mathrm {sgn}\relax (x)}\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccosh(d*x^2-1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 37, normalized size = 1.12 \[ a x +b \left (x \,\mathrm {arccosh}\left (d \,x^{2}-1\right )-\frac {2 x \sqrt {d \,x^{2}-2}}{\sqrt {d \,x^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arccosh(d*x^2-1),x)

[Out]

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

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maxima [A]  time = 0.65, size = 44, normalized size = 1.33 \[ {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccosh(d*x^2-1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.53, size = 32, normalized size = 0.97 \[ a\,x+b\,x\,\mathrm {acosh}\left (d\,x^2-1\right )-\frac {2\,b\,\mathrm {sign}\relax (x)\,\sqrt {d\,x^2-2}}{\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*acosh(d*x^2 - 1),x)

[Out]

a*x + b*x*acosh(d*x^2 - 1) - (2*b*sign(x)*(d*x^2 - 2)^(1/2))/d^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*acosh(d*x**2-1),x)

[Out]

Integral(a + b*acosh(d*x**2 - 1), x)

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