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 = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 12
Rule 74
Rule 460
Rule 5680
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.45, size = 83, normalized size = 0.97 \[ -\frac {3 \, a c^{3} d x^{3} - 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} - 3 \, b c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d x^{2} - 7 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.85 \[ \frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \mathrm {arccosh}\left (c x \right )}{3}-c x \,\mathrm {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 97, normalized size = 1.13 \[ -\frac {1}{3} \, a c^{2} d x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 97, normalized size = 1.13 \[ \begin {cases} - \frac {a c^{2} d x^{3}}{3} + a d x - \frac {b c^{2} d x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {b c d x^{2} \sqrt {c^{2} x^{2} - 1}}{9} + b d x \operatorname {acosh}{\left (c x \right )} - \frac {7 b d \sqrt {c^{2} x^{2} - 1}}{9 c} & \text {for}\: c \neq 0 \\d x \left (a + \frac {i \pi b}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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