Optimal. Leaf size=30 \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c}+b x \text{csch}^{-1}(c x) \]
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Rubi [A] time = 0.0209549, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6278, 266, 63, 208} \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c}+b x \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6278
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=a x+b \int \text{csch}^{-1}(c x) \, dx\\ &=a x+b x \text{csch}^{-1}(c x)+\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \text{csch}^{-1}(c x)-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c}\\ &=a x+b x \text{csch}^{-1}(c x)-(b c) \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )\\ &=a x+b x \text{csch}^{-1}(c x)+\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0595727, size = 44, normalized size = 1.47 \[ a x+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b x \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.198, size = 36, normalized size = 1.2 \begin{align*} ax+bx{\rm arccsch} \left (cx\right )+{\frac{b}{c}\ln \left ( cx+cx\sqrt{1+{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01569, size = 66, normalized size = 2.2 \begin{align*} a x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25656, size = 320, normalized size = 10.67 \begin{align*} \frac{a c x + b c \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - b c \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - b \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) +{\left (b c x - b c\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c} \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 \left (a + b \operatorname{acsch}{\left (c x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int b \operatorname{arcsch}\left (c x\right ) + a\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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