3.3.57 \(\int \frac {1}{\sqrt {a+b \cosh ^{-1}(1+d x^2)}} \, dx\) [257]

Optimal. Leaf size=165 \[ \frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt {b} d x}+\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt {b} d x} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 165, 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 = {6004} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\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}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6004

Int[1/Sqrt[(a_.) + ArcCosh[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b
)])*Sinh[ArcCosh[1 + d*x^2]/2]*(Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)), x] + Simp[Sqrt[
Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[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}} \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 ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt {b} d x}+\frac {\sqrt {\frac {\pi }{2}} \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 ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt {b} d x}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 166, normalized size = 1.01 \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} x \left (\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 )+\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 )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{\sqrt {b} \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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

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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