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

Optimal. Leaf size=206 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {2 \cosh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]

[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))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/x-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))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/x+2*cosh(1/2*arccosh(d*x^2-1))^2*(a+b*arcc
osh(d*x^2-1))^(1/2)/d/x

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 206, 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 = {5879} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {2 \cosh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5879

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 178, normalized size = 0.86 \[ \frac {\cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (-\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )+4 \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}\right )}{2 d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

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((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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(d*x^2 - 1))^(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________