Optimal. Leaf size=64 \[ -\frac{2 b \tan ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{a+b x-1}}\right )}{\sqrt{1-a^2}}-\frac{\cosh ^{-1}(a+b x)}{x} \]
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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.
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Rule 5866
Rule 5802
Rule 93
Rule 205
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.
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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.
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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.
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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.
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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.
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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.
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