Optimal. Leaf size=83 \[ \frac{2 c \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e \sqrt{c d-e} \sqrt{c d+e}}-\frac{\cosh ^{-1}(c x)}{e (d+e x)} \]
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Rubi [A] time = 0.0877095, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5802, 93, 208} \[ \frac{2 c \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e \sqrt{c d-e} \sqrt{c d+e}}-\frac{\cosh ^{-1}(c x)}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 5802
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac{\cosh ^{-1}(c x)}{e (d+e x)}+\frac{c \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)} \, dx}{e}\\ &=-\frac{\cosh ^{-1}(c x)}{e (d+e x)}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{c d-e-(c d+e) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{e}\\ &=-\frac{\cosh ^{-1}(c x)}{e (d+e x)}+\frac{2 c \tanh ^{-1}\left (\frac{\sqrt{c d+e} \sqrt{1+c x}}{\sqrt{c d-e} \sqrt{-1+c x}}\right )}{\sqrt{c d-e} e \sqrt{c d+e}}\\ \end{align*}
Mathematica [A] time = 0.0953047, size = 92, normalized size = 1.11 \[ \frac{\frac{c \left (\log (d+e x)-\log \left (-\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d^2-e^2}+c^2 d x+e\right )\right )}{\sqrt{c^2 d^2-e^2}}-\frac{\cosh ^{-1}(c x)}{d+e x}}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 126, normalized size = 1.5 \begin{align*} -{\frac{c{\rm arccosh} \left (cx\right )}{ \left ( cxe+cd \right ) e}}-{\frac{c}{{e}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( -2\,{\frac{1}{cxe+cd} \left ({c}^{2}dx-\sqrt{{c}^{2}{x}^{2}-1}\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}e+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}{\frac{1}{\sqrt{{c}^{2}{x}^{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 = 2.31234, size = 903, normalized size = 10.88 \begin{align*} \left [\frac{{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + \sqrt{c^{2} d^{2} - e^{2}}{\left (c d e x + c d^{2}\right )} \log \left (\frac{c^{3} d^{2} x + c d e + \sqrt{c^{2} d^{2} - e^{2}}{\left (c^{2} d x + e\right )} +{\left (c^{2} d^{2} + \sqrt{c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{e x + d}\right ) +{\left (c^{2} d^{3} - d e^{2} +{\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, \frac{{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 2 \, \sqrt{-c^{2} d^{2} + e^{2}}{\left (c d e x + c d^{2}\right )} \arctan \left (-\frac{\sqrt{-c^{2} d^{2} + e^{2}} \sqrt{c^{2} x^{2} - 1} e - \sqrt{-c^{2} d^{2} + e^{2}}{\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) +{\left (c^{2} d^{3} - d e^{2} +{\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\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 (c x \right )}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38854, size = 286, normalized size = 3.45 \begin{align*}{\left (\frac{e^{\left (-1\right )} \log \left ({\left | -c^{2} d + \sqrt{c^{2} d^{2} - e^{2}}{\left | c \right |} \right |}\right ) \mathrm{sgn}\left (\frac{1}{x e + d}\right )}{\sqrt{c^{2} d^{2} - e^{2}}} - \frac{e^{\left (-1\right )} \log \left ({\left | -c^{2} d + \sqrt{c^{2} d^{2} - e^{2}}{\left (\sqrt{c^{2} - \frac{2 \, c^{2} d}{x e + d} + \frac{c^{2} d^{2}}{{\left (x e + d\right )}^{2}} - \frac{e^{2}}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c^{2} d^{2} e^{2} - e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{\sqrt{c^{2} d^{2} - e^{2}} \mathrm{sgn}\left (\frac{1}{x e + d}\right )}\right )} c - \frac{e^{\left (-1\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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