Optimal. Leaf size=57 \[ -\frac{b \tanh ^{-1}\left (\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right )}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.0704265, antiderivative size = 57, 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 = {5865, 5801, 725, 206} \[ -\frac{b \tanh ^{-1}\left (\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right )}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5801
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a+b x)}{x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sinh ^{-1}(a+b x)}{x}+\operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+\frac{x}{b}\right ) \sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac{\sinh ^{-1}(a+b x)}{x}-\operatorname{Subst}\left (\int \frac{1}{\frac{1}{b^2}+\frac{a^2}{b^2}-x^2} \, dx,x,\frac{\frac{1}{b}+\frac{a (a+b x)}{b}}{\sqrt{1+(a+b x)^2}}\right )\\ &=-\frac{\sinh ^{-1}(a+b x)}{x}-\frac{b \tanh ^{-1}\left (\frac{b \left (\frac{1}{b}+\frac{a (a+b x)}{b}\right )}{\sqrt{1+a^2} \sqrt{1+(a+b x)^2}}\right )}{\sqrt{1+a^2}}\\ \end{align*}
Mathematica [A] time = 0.0385993, size = 57, normalized size = 1. \[ -\frac{b \tanh ^{-1}\left (\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right )}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 71, normalized size = 1.3 \begin{align*} -{\frac{{\it Arcsinh} \left ( bx+a \right ) }{x}}-{b\ln \left ({\frac{1}{bx} \left ( 2\,{a}^{2}+2+2\,xab+2\,\sqrt{{a}^{2}+1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{{a}^{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 = 2.77731, size = 405, normalized size = 7.11 \begin{align*} \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} \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{asinh}{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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