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

Optimal. Leaf size=166 \[ \frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}-\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x} \]

[Out]

1/2*cosh(1/2*arccosh(d*x^2-1))*erfi(1/2*(a+b*arccosh(d*x^2-1))^(1/2)*2^(1/2)/b^(1/2))*(cosh(1/2*a/b)-sinh(1/2*
a/b))*2^(1/2)*Pi^(1/2)/d/x/b^(1/2)-1/2*cosh(1/2*arccosh(d*x^2-1))*erf(1/2*(a+b*arccosh(d*x^2-1))^(1/2)*2^(1/2)
/b^(1/2))*(cosh(1/2*a/b)+sinh(1/2*a/b))*2^(1/2)*Pi^(1/2)/d/x/b^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 = {5884} \[ \frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}-\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)]))/(Sqrt[b]*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)]))/(Sqrt[b]*d*x)

Rule 5884

Int[1/Sqrt[(a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> 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]])/(Sqrt[b]*d*x), x] - Simp[(S
qrt[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]])/(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}} \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 )}{\sqrt {b} 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 )}{\sqrt {b} d x}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 134, normalized size = 0.81 \[ -\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )+\left (\sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )\right )}{\sqrt {b} d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]index.cc index_m i_lex_is_greater Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argumen
t Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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