3.88 \(\int \frac{\cosh ^{-1}(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{2 b \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{a+b x-1}}\right )}{\sqrt{1-a^2}}-\frac{\cosh ^{-1}(a+b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0828594, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5866, 5802, 93, 205} \[ -\frac{2 b \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{a+b x-1}}\right )}{\sqrt{1-a^2}}-\frac{\cosh ^{-1}(a+b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a + b*x]/x^2,x]

[Out]

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

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5802

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcCosh[c*x
])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.107167, size = 83, normalized size = 1.3 \[ -\frac{\cosh ^{-1}(a+b x)}{x}-\frac{i b \log \left (\frac{2 \left (\sqrt{a+b x-1} \sqrt{a+b x+1}+\frac{i \left (a^2+a b x-1\right )}{\sqrt{1-a^2}}\right )}{b x}\right )}{\sqrt{1-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a + b*x]/x^2,x]

[Out]

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

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Maple [A]  time = 0.014, size = 97, normalized size = 1.5 \begin{align*} -{\frac{{\rm arccosh} \left (bx+a\right )}{x}}-{\frac{b}{ \left ( a-1 \right ) \left ( 1+a \right ) }\sqrt{bx+a-1}\sqrt{bx+a+1}\sqrt{{a}^{2}-1}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}-1}\sqrt{ \left ( bx+a \right ) ^{2}-1}+a \left ( bx+a \right ) -1}{bx}} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arccosh(b*x+a)/x-b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(a^2-1)^(1/2)*ln(2*((a^2-1)^(1/2)*((b*x+a)^2-1)^(1/2)+a*(b
*x+a)-1)/b/x)/((b*x+a)^2-1)^(1/2)/(a-1)/(1+a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.6116, size = 771, normalized size = 12.05 \begin{align*} \left [\frac{\sqrt{a^{2} - 1} b x \log \left (\frac{a^{2} b x + a^{3} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{2} - \sqrt{a^{2} - 1} a - 1\right )} -{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} - a}{x}\right ) +{\left (a^{2} - 1\right )} x \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) -{\left (a^{2} -{\left (a^{2} - 1\right )} x - 1\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{{\left (a^{2} - 1\right )} x}, \frac{2 \, \sqrt{-a^{2} + 1} b x \arctan \left (-\frac{\sqrt{-a^{2} + 1} b x - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt{-a^{2} + 1}}{a^{2} - 1}\right ) +{\left (a^{2} - 1\right )} x \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) -{\left (a^{2} -{\left (a^{2} - 1\right )} x - 1\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{{\left (a^{2} - 1\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(a^2 - 1)*b*x*log((a^2*b*x + a^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - sqrt(a^2 - 1)*a - 1) - (a*b*
x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) + (a^2 - 1)*x*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (a^2 - (a
^2 - 1)*x - 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)))/((a^2 - 1)*x), (2*sqrt(-a^2 + 1)*b*x*arctan(-
(sqrt(-a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqrt(-a^2 + 1))/(a^2 - 1)) + (a^2 - 1)*x*log(-b*x - a
+ sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (a^2 - (a^2 - 1)*x - 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)
))/((a^2 - 1)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a + b*x)/x**2, x)

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Giac [A]  time = 1.20085, size = 99, normalized size = 1.55 \begin{align*} \frac{2 \, b \arctan \left (-\frac{x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt{-a^{2} + 1}}\right )}{\sqrt{-a^{2} + 1}} - \frac{\log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} - 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*arctan(-(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/sqrt(-a^2 + 1))/sqrt(-a^2 + 1) - log(b*x + a + sqrt
((b*x + a)^2 - 1))/x