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

Optimal. Leaf size=206 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]

[Out]

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

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Rubi [A]  time = 0.0263156, antiderivative size = 206, 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 = {5879} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5879

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

Rubi steps

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

Mathematica [A]  time = 0.250142, size = 178, normalized size = 0.86 \[ \frac{\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (-\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{a}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+4 \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}\right )}{2 d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[ArcCosh[-1 + d*x^2]/2]*(4*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[ArcCosh[-1 + d*x^2]/2] + Sqrt[b]*Sqrt[2*P
i]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(-Cosh[a/(2*b)] + Sinh[a/(2*b)]) - Sqrt[b]*Sqrt[2*P
i]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])))/(2*d*x)

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*arccosh(d*x^2-1))^(1/2),x)

[Out]

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

[Out]

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

[Out]

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

[Out]

Exception raised: TypeError