Optimal. Leaf size=73 \[ 8 b^2 x+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2 \]
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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 = {6001, 8}
\begin {gather*} 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 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 6001
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 = 0.04, size = 104, normalized size = 1.42 \begin {gather*} \left (a^2+8 b^2\right ) x-\frac {4 a b \sqrt {d x^2} \sqrt {-2+d x^2}}{d x}+\frac {2 b \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \cosh ^{-1}\left (-1+d x^2\right )}{d x}+b^2 x \cosh ^{-1}\left (-1+d x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 = 0.35, size = 128, normalized size = 1.75 \begin {gather*} 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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 131, normalized size = 1.79 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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