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

Optimal. Leaf size=49 \[ a x-\frac{2 b \sqrt{\frac{d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+b x \cosh ^{-1}\left (d x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0390155, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5901, 12, 6719, 261} \[ a x-\frac{2 b \sqrt{\frac{d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+b x \cosh ^{-1}\left (d x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCosh[1 + d*x^2],x]

[Out]

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

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]

Rule 12

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 261

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

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

Mathematica [A]  time = 0.0588238, size = 37, normalized size = 0.76 \[ a x-\frac{2 b x}{\sqrt{\frac{d x^2}{d x^2+2}}}+b x \cosh ^{-1}\left (d x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCosh[1 + d*x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 0.8 \begin{align*} ax+b \left ( x{\rm arccosh} \left (d{x}^{2}+1\right )-2\,{\frac{x\sqrt{d{x}^{2}+2}}{\sqrt{d{x}^{2}}}} \right ) \end{align*}

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 = 1.03273, size = 59, normalized size = 1.2 \begin{align*}{\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 \end{align*}

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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Fricas [A]  time = 1.98192, size = 132, normalized size = 2.69 \begin{align*} \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} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (d x^{2} + 1 \right )}\right )\, dx \end{align*}

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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Giac [A]  time = 1.12393, size = 88, normalized size = 1.8 \begin{align*}{\left (2 \, d{\left (\frac{\sqrt{2} \mathrm{sgn}\left (x\right )}{d^{\frac{3}{2}}} - \frac{\sqrt{d^{2} x^{2} + 2 \, d}}{d^{2} \mathrm{sgn}\left (x\right )}\right )} + x \log \left (d x^{2} + \sqrt{{\left (d x^{2} + 1\right )}^{2} - 1} + 1\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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