Optimal. Leaf size=86 \[ -\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} b c d x^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]
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Rubi [A] time = 0.074217, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5680, 12, 460, 74} \[ -\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} b c d x^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]
Antiderivative was successfully verified.
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Rule 5680
Rule 12
Rule 460
Rule 74
Rubi steps
\begin{align*} \int \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c d) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{9} b c d x^2 \sqrt{-1+c x} \sqrt{1+c x}+d x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} (7 b c d) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{7 b d \sqrt{-1+c x} \sqrt{1+c x}}{9 c}+\frac{1}{9} b c d x^2 \sqrt{-1+c x} \sqrt{1+c x}+d x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0845899, size = 71, normalized size = 0.83 \[ \frac{d \left (a \left (9 c x-3 c^3 x^3\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-7\right )-3 b c x \left (c^2 x^2-3\right ) \cosh ^{-1}(c x)\right )}{9 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 73, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ( -da \left ({\frac{{c}^{3}{x}^{3}}{3}}-cx \right ) -db \left ({\frac{{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )}{3}}-cx{\rm arccosh} \left (cx\right )-{\frac{{c}^{2}{x}^{2}-7}{9}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18129, size = 131, normalized size = 1.52 \begin{align*} -\frac{1}{3} \, a c^{2} d x^{3} - \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82291, size = 185, normalized size = 2.15 \begin{align*} -\frac{3 \, a c^{3} d x^{3} - 9 \, a c d x + 3 \,{\left (b c^{3} d x^{3} - 3 \, b c d x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c^{2} d x^{2} - 7 \, b d\right )} \sqrt{c^{2} x^{2} - 1}}{9 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.755015, size = 97, normalized size = 1.13 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{3}}{3} + a d x - \frac{b c^{2} d x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b c d x^{2} \sqrt{c^{2} x^{2} - 1}}{9} + b d x \operatorname{acosh}{\left (c x \right )} - \frac{7 b d \sqrt{c^{2} x^{2} - 1}}{9 c} & \text{for}\: c \neq 0 \\d x \left (a + \frac{i \pi b}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37603, size = 151, normalized size = 1.76 \begin{align*} -\frac{1}{3} \, a c^{2} d x^{3} - \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b c^{2} d +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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