Optimal. Leaf size=33 \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5901, 12, 191} \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 5901
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right ) \, dx &=a x+b \int \cosh ^{-1}\left (-1+d x^2\right ) \, dx\\ &=a x+b x \cosh ^{-1}\left (-1+d x^2\right )-b \int \frac {2}{\sqrt {1-\frac {2}{d x^2}}} \, dx\\ &=a x+b x \cosh ^{-1}\left (-1+d x^2\right )-(2 b) \int \frac {1}{\sqrt {1-\frac {2}{d x^2}}} \, dx\\ &=a x-2 b \sqrt {1-\frac {2}{d x^2}} x+b x \cosh ^{-1}\left (-1+d x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.00 \[ a x-2 b x \sqrt {1-\frac {2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 63, normalized size = 1.91 \[ \frac {b d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) + a d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 67, normalized size = 2.03 \[ {\left (x \log \left (d x^{2} + \sqrt {{\left (d x^{2} - 1\right )}^{2} - 1} - 1\right ) + \frac {2 \, \sqrt {2} \sqrt {-d} \mathrm {sgn}\relax (x)}{d} - \frac {2 \, \sqrt {d^{2} x^{2} - 2 \, d}}{d \mathrm {sgn}\relax (x)}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 1.12 \[ a x +b \left (x \,\mathrm {arccosh}\left (d \,x^{2}-1\right )-\frac {2 x \sqrt {d \,x^{2}-2}}{\sqrt {d \,x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 44, normalized size = 1.33 \[ {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 32, normalized size = 0.97 \[ a\,x+b\,x\,\mathrm {acosh}\left (d\,x^2-1\right )-\frac {2\,b\,\mathrm {sign}\relax (x)\,\sqrt {d\,x^2-2}}{\sqrt {d}} \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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