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

Optimal. Leaf size=98 \[ \frac{x \cosh \left (\frac{a}{2 b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \sinh \left (\frac{a}{2 b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0153676, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5882} \[ \frac{x \cosh \left (\frac{a}{2 b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \sinh \left (\frac{a}{2 b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5882

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

Rubi steps

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

Mathematica [A]  time = 0.129587, size = 86, normalized size = 0.88 \[ -\frac{\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )\right )}{b d x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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)),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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)),x, algorithm="giac")

[Out]

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