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 = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 93
Rule 208
Rule 5802
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*}
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Mathematica [A] time = 0.10, 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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fricas [B] time = 0.56, size = 454, normalized size = 5.47 \[ \left [\frac {{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} d^{2} - e^{2}} {\left (c d e x + c d^{2}\right )} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + {\left (c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, \frac {{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c d e x + c d^{2}\right )} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\left (c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 215, normalized size = 2.59 \[ {\left (\frac {c e^{3} \log \left ({\left | c^{2} d - \sqrt {c^{2} d^{2} - e^{2}} {\left | c \right |} \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c^{2} d^{2} - e^{2}}} - \frac {c e^{3} \log \left ({\left | c^{2} d - \sqrt {c^{2} d^{2} - e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{x e + d} + \frac {c^{2} d^{2}}{{\left (x e + d\right )}^{2}} - \frac {e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} - e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{\sqrt {c^{2} d^{2} - e^{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-4\right )} - \frac {e^{\left (-1\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{x e + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 126, normalized size = 1.52 \[ -\frac {c \,\mathrm {arccosh}\left (c x \right )}{\left (c x e +c d \right ) e}-\frac {c \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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