Optimal. Leaf size=73 \[ x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+8 b^2 x \]
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Rubi [A] time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5880, 8} \[ \frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+8 b^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 5880
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2 \, dx &=\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2+\left (8 b^2\right ) \int 1 \, dx\\ &=8 b^2 x+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.07, size = 104, normalized size = 1.42 \[ x \left (a^2+8 b^2\right )-\frac {4 a b \sqrt {d x^2} \sqrt {d x^2-2}}{d x}+\frac {2 b \cosh ^{-1}\left (d x^2-1\right ) \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2-2}\right )}{d x}+b^2 x \cosh ^{-1}\left (d x^2-1\right )^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 131, normalized size = 1.79 \[ \frac {b^{2} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b + 2 \, {\left (a b d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )}{d x} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 128, normalized size = 1.75 \[ b^{2} x \operatorname {arcosh}\left (d x^{2} - 1\right )^{2} + 4 \, b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d x^{2}} - 1\right )}{\sqrt {d x^{2} - 2} d^{2}}\right )} + 2 \, {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} a b + a^{2} x \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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