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

Optimal. Leaf size=205 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {2 \sinh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x} \]

[Out]

-1/2*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))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/x+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)+s
inh(1/2*a/b))*sinh(1/2*arccosh(d*x^2+1))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/x+2*sinh(1/2*arccosh(d*x^2+1))^2*(a+b*arcc
osh(d*x^2+1))^(1/2)/d/x

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Rubi [A]  time = 0.03, antiderivative size = 205, 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 = {5878} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {2 \sinh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*ArcCosh[1 + d*x^2]],x]

[Out]

-((Sqrt[b]*Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*S
inh[ArcCosh[1 + d*x^2]/2])/(d*x)) + (Sqrt[b]*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])/(d*x) + (2*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Sinh[Arc
Cosh[1 + d*x^2]/2]^2)/(d*x)

Rule 5878

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

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Mathematica [A]  time = 0.31, size = 210, normalized size = 1.02 \[ \frac {x \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )+4 \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}\right )}{2 \sqrt {d x^2} \sqrt {\frac {d x^2}{d x^2+2}} \sqrt {d x^2+2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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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((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((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 \[ \int \sqrt {a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )}\, dx \]

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)

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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)

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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)

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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)

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