Optimal. Leaf size=97 \[ -\frac{1}{4} \left (\frac{e}{c^2}+\frac{2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac{\sqrt{c x-1} \sqrt{c x+1} (d+e x)}{4 c}+\frac{\cosh ^{-1}(c x) (d+e x)^2}{2 e}-\frac{3 d \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]
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Rubi [A] time = 0.0417405, antiderivative size = 97, 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 = {5802, 90, 80, 52} \[ -\frac{1}{4} \left (\frac{e}{c^2}+\frac{2 d^2}{e}\right ) \cosh ^{-1}(c x)-\frac{\sqrt{c x-1} \sqrt{c x+1} (d+e x)}{4 c}+\frac{\cosh ^{-1}(c x) (d+e x)^2}{2 e}-\frac{3 d \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]
Antiderivative was successfully verified.
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Rule 5802
Rule 90
Rule 80
Rule 52
Rubi steps
\begin{align*} \int (d+e x) \cosh ^{-1}(c x) \, dx &=\frac{(d+e x)^2 \cosh ^{-1}(c x)}{2 e}-\frac{c \int \frac{(d+e x)^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}\\ &=-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}+\frac{(d+e x)^2 \cosh ^{-1}(c x)}{2 e}-\frac{\int \frac{2 c^2 d^2+e^2+3 c^2 d e x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c e}\\ &=-\frac{3 d \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}+\frac{(d+e x)^2 \cosh ^{-1}(c x)}{2 e}-\frac{1}{4} \left (\frac{2 c d^2}{e}+\frac{e}{c}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{3 d \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)}{4 c}-\frac{1}{4} \left (\frac{2 d^2}{e}+\frac{e}{c^2}\right ) \cosh ^{-1}(c x)+\frac{(d+e x)^2 \cosh ^{-1}(c x)}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0731214, size = 73, normalized size = 0.75 \[ -\frac{-2 c^2 x \cosh ^{-1}(c x) (2 d+e x)+c \sqrt{c x-1} \sqrt{c x+1} (4 d+e x)+2 e \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{4 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.012, size = 107, normalized size = 1.1 \begin{align*}{\frac{{\rm arccosh} \left (cx\right ){x}^{2}e}{2}}+{\rm arccosh} \left (cx\right )xd-{\frac{ex}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{d}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{e}{4\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\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 [A] time = 1.16758, size = 123, normalized size = 1.27 \begin{align*} -\frac{1}{4} \, c{\left (\frac{\sqrt{c^{2} x^{2} - 1} e x}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} d}{c^{2}} + \frac{e \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )} + \frac{1}{2} \,{\left (e x^{2} + 2 \, d x\right )} \operatorname{arcosh}\left (c x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02282, size = 143, normalized size = 1.47 \begin{align*} \frac{{\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \sqrt{c^{2} x^{2} - 1}{\left (c e x + 4 \, c d\right )}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.367506, size = 80, normalized size = 0.82 \begin{align*} \begin{cases} d x \operatorname{acosh}{\left (c x \right )} + \frac{e x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{d \sqrt{c^{2} x^{2} - 1}}{c} - \frac{e x \sqrt{c^{2} x^{2} - 1}}{4 c} - \frac{e \operatorname{acosh}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\\frac{i \pi \left (d x + \frac{e x^{2}}{2}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17712, size = 117, normalized size = 1.21 \begin{align*} \frac{1}{2} \,{\left (x^{2} e + 2 \, d x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{1}{4} \, \sqrt{c^{2} x^{2} - 1}{\left (\frac{x e}{c} + \frac{4 \, d}{c}\right )} + \frac{e \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{4 \, c{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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