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]

1/2*x*cosh(1/2*a/b)*Shi(1/2*(a+b*arccosh(d*x^2-1))/b)/b*2^(1/2)/(d*x^2)^(1/2)-1/2*x*Chi(1/2*(a+b*arccosh(d*x^2
-1))/b)*sinh(1/2*a/b)/b*2^(1/2)/(d*x^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.14, 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))

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a}\,{d x} \]

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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{a+b\,\mathrm {acosh}\left (d\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acosh(d*x^2 - 1)),x)

[Out]

int(1/(a + b*acosh(d*x^2 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}}\, dx \]

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)

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