3.3.45 \(\int \frac {1}{(a+b \cosh ^{-1}(1+d x^2))^2} \, dx\) [245]

Optimal. Leaf size=150 \[ -\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{2 b d x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}-\frac {x \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}} \]

[Out]

1/4*x*cosh(1/2*a/b)*Shi(1/2*(a+b*arccosh(d*x^2+1))/b)/b^2*2^(1/2)/(d*x^2)^(1/2)-1/4*x*Chi(1/2*(a+b*arccosh(d*x
^2+1))/b)*sinh(1/2*a/b)/b^2*2^(1/2)/(d*x^2)^(1/2)-1/2*(d*x^2)^(1/2)*(d*x^2+2)^(1/2)/b/d/x/(a+b*arccosh(d*x^2+1
))

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Rubi [A]
time = 0.02, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6008} \begin {gather*} -\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {\sqrt {d x^2} \sqrt {d x^2+2}}{2 b d x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6008

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

Rubi steps

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

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Mathematica [A]
time = 0.63, size = 130, normalized size = 0.87 \begin {gather*} -\frac {\frac {2 b \sqrt {d x^2} \sqrt {2+d x^2}}{a d+b d \cosh ^{-1}\left (1+d x^2\right )}+x^2 \text {csch}\left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right ) \left (\text {Chi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (1+d x^2\right )}{2 b}\right )\right )}{4 b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((2*b*Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(a*d + b*d*ArcCosh[1 + d*x^2]) + x^2*Csch[ArcCosh[1 + d*x^2]/2]*(CoshI
ntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]*Sinh[a/(2*b)] - Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2]
)/(2*b)]))/(b^2*x)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*(d^2*x^4 + 3*d*x^2 + (d^(3/2)*x^3 + 2*sqrt(d)*x)*sqrt(d*x^2 + 2) + 2)/(a*b*d^2*x^3 + 2*a*b*d*x + (b^2*d^2
*x^3 + 2*b^2*d*x + (b^2*d^(3/2)*x^2 + b^2*sqrt(d))*sqrt(d*x^2 + 2))*log(d*x^2 + sqrt(d*x^2 + 2)*sqrt(d)*x + 1)
+ (a*b*d^(3/2)*x^2 + a*b*sqrt(d))*sqrt(d*x^2 + 2)) + integrate(1/2*(d^3*x^6 + 3*d^2*x^4 + (d^2*x^4 + d*x^2 +
2)*(d*x^2 + 2) + (2*d^(5/2)*x^5 + 4*d^(3/2)*x^3 + sqrt(d)*x)*sqrt(d*x^2 + 2) - 4)/(a*b*d^3*x^6 + 4*a*b*d^2*x^4
+ 4*a*b*d*x^2 + (a*b*d^2*x^4 + 2*a*b*d*x^2 + a*b)*(d*x^2 + 2) + (b^2*d^3*x^6 + 4*b^2*d^2*x^4 + 4*b^2*d*x^2 +
(b^2*d^2*x^4 + 2*b^2*d*x^2 + b^2)*(d*x^2 + 2) + 2*(b^2*d^(5/2)*x^5 + 3*b^2*d^(3/2)*x^3 + 2*b^2*sqrt(d)*x)*sqrt
(d*x^2 + 2))*log(d*x^2 + sqrt(d*x^2 + 2)*sqrt(d)*x + 1) + 2*(a*b*d^(5/2)*x^5 + 3*a*b*d^(3/2)*x^3 + 2*a*b*sqrt(
d)*x)*sqrt(d*x^2 + 2)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acosh(d*x^2 + 1))^2,x)

[Out]

int(1/(a + b*acosh(d*x^2 + 1))^2, x)

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