3.265 \(\int \frac{1}{(a+b \cosh ^{-1}(-1+d x^2))^{3/2}} \, dx\)

Optimal. Leaf size=212 \[ \frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{b^{3/2} d x}+\frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{b^{3/2} d x}-\frac{\sqrt{d x^2} \sqrt{d x^2-2}}{b d x \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}} \]

[Out]

-((Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])) + (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)]))/(b^(3/2)*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)]))/(b^(3/2)*d*x)

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Rubi [A]  time = 0.0273975, antiderivative size = 212, 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 = {5886} \[ \frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{b^{3/2} d x}+\frac{\sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{b^{3/2} d x}-\frac{\sqrt{d x^2} \sqrt{d x^2-2}}{b d x \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^(-3/2),x]

[Out]

-((Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])) + (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)]))/(b^(3/2)*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)]))/(b^(3/2)*d*x)

Rule 5886

Int[((a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.))^(-3/2), x_Symbol] :> -Simp[(Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*
x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]), x] + (Simp[(Sqrt[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]])/(b^(3/2)*d*x), x] + 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]])/(b^(3/2)*d*x), x]
) /; FreeQ[{a, b, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}} \, dx &=-\frac{\sqrt{d x^2} \sqrt{-2+d x^2}}{b d x \sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )}}+\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 )}{b^{3/2} 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 )}{b^{3/2} d x}\\ \end{align*}

Mathematica [A]  time = 0.973442, size = 209, normalized size = 0.99 \[ \frac{\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sqrt{2 \pi } \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )+\sqrt{2 \pi } \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )-4 \sqrt{b} \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )\right )}{2 b^{3/2} d x \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[-1 + d*x^2])^(-3/2),x]

[Out]

(Cosh[ArcCosh[-1 + d*x^2]/2]*(Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/
(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]) + Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b
*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]) - 4*Sqrt[b]*Sinh[ArcCosh[-1 + d*x^2]/
2]))/(2*b^(3/2)*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}-1\right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccosh(d*x^2-1))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x^2 - 1) + a)^(-3/2), 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))^(3/2),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}{\left (a + b \operatorname{acosh}{\left (d x^{2} - 1 \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acosh(d*x**2-1))**(3/2),x)

[Out]

Integral((a + b*acosh(d*x**2 - 1))**(-3/2), 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))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError