Optimal. Leaf size=132 \[ -\frac{c \sqrt{c x-1} \sqrt{c x+1}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{c^3 d \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e (c d-e)^{3/2} (c d+e)^{3/2}}-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.134899, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5802, 96, 93, 208} \[ -\frac{c \sqrt{c x-1} \sqrt{c x+1}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{c^3 d \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e (c d-e)^{3/2} (c d+e)^{3/2}}-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 5802
Rule 96
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac{c \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2} \, dx}{2 e}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac{\left (c^3 d\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)} \, dx}{2 e \left (c^2 d^2-e^2\right )}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac{\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac{c^3 d \tanh ^{-1}\left (\frac{\sqrt{c d+e} \sqrt{1+c x}}{\sqrt{c d-e} \sqrt{-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.185377, size = 190, normalized size = 1.44 \[ \frac{c (d+e x) \left (-e \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d^2-e^2}-c^2 d (d+e x) \log \left (-\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d^2-e^2}+c^2 d x+e\right )+c^2 d (d+e x) \log (d+e x)\right )-\left (c^2 d^2-e^2\right )^{3/2} \cosh ^{-1}(c x)}{2 e (c d-e) (c d+e) \sqrt{c^2 d^2-e^2} (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 338, normalized size = 2.6 \begin{align*} -{\frac{{c}^{2}{\rm arccosh} \left (cx\right )}{2\, \left ( cxe+cd \right ) ^{2}e}}-{\frac{{c}^{4}xd}{2\,e \left ( cd+e \right ) \left ( cd-e \right ) \left ( cxe+cd \right ) }\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{{c}^{2}{x}^{2}-1}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{2}{c}^{4}}{2\,{e}^{2} \left ( cd+e \right ) \left ( cd-e \right ) \left ( cxe+cd \right ) }\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{{c}^{2}{x}^{2}-1}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}}-{\frac{{c}^{2}}{ \left ( 2\,cd+2\,e \right ) \left ( cd-e \right ) \left ( cxe+cd \right ) }\sqrt{cx-1}\sqrt{cx+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 = 2.959, size = 2040, normalized size = 15.45 \begin{align*} \text{result too large to display} \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 )^{3}}\, 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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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