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

Optimal. Leaf size=166 \[ \frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x}-\frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x} \]

[Out]

(Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)]))/(Sqrt[b]*d*x) - (Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]
/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Sqrt[b]*d*x)

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Rubi [A]  time = 0.025722, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5884} \[ \frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x}-\frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*ArcCosh[-1 + d*x^2]],x]

[Out]

(Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)]))/(Sqrt[b]*d*x) - (Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]
/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Sqrt[b]*d*x)

Rule 5884

Int[1/Sqrt[(a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2
*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] - Simp[(S
qrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt
[2*b]])/(Sqrt[b]*d*x), x] /; FreeQ[{a, b, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.254311, size = 134, normalized size = 0.81 \[ -\frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+\left (\sinh \left (\frac{a}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )\right )}{\sqrt{b} d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*(Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(-Cosh[a/(2
*b)] + Sinh[a/(2*b)]) + Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])
))/(Sqrt[b]*d*x))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+b{\rm arccosh} \left (d{x}^{2}-1\right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \operatorname{arcosh}\left (d x^{2} - 1\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*acosh(d*x**2 - 1)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError