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

Optimal. Leaf size=165 \[ \frac{\sqrt{\frac{\pi }{2}} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x}+\frac{\sqrt{\frac{\pi }{2}} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x} \]

[Out]

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

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Rubi [A]  time = 0.0245432, antiderivative size = 165, 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 = {5883} \[ \frac{\sqrt{\frac{\pi }{2}} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x}+\frac{\sqrt{\frac{\pi }{2}} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b} d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5883

Int[1/Sqrt[(a_.) + ArcCosh[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*
b)])*Sinh[ArcCosh[1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]])/(Sqrt[b]*d*x), x] + Simp[(Sqrt
[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[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}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt{b} d x}+\frac{\sqrt{\frac{\pi }{2}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt{b} d x}\\ \end{align*}

Mathematica [A]  time = 0.304812, size = 166, normalized size = 1.01 \[ \frac{\sqrt{\frac{\pi }{2}} x \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (\left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+\left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )\right )}{\sqrt{b} \sqrt{d x^2} \sqrt{\frac{d x^2}{d x^2+2}} \sqrt{d x^2+2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.067, 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