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]

-((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)

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Rubi [A]  time = 0.0264593, antiderivative size = 205, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.291669, 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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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b{\rm arccosh} \left (d{x}^{2}+1\right )}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{arcosh}\left (d x^{2} + 1\right ) + a}\,{d x} \end{align*}

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

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: UnboundLocalError

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

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

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