Optimal. Leaf size=110 \[ 24 a b^2 x+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^3+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}-48 b^3 x \sqrt {1-\frac {2}{d x^2}}+24 b^3 x \cosh ^{-1}\left (d x^2-1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5880, 5901, 12, 191} \[ 24 a b^2 x+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^3-48 b^3 x \sqrt {1-\frac {2}{d x^2}}+24 b^3 x \cosh ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 5880
Rule 5901
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3 \, dx &=\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right ) \, dx\\ &=24 a b^2 x+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3+\left (24 b^3\right ) \int \cosh ^{-1}\left (-1+d x^2\right ) \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3-\left (24 b^3\right ) \int \frac {2}{\sqrt {1-\frac {2}{d x^2}}} \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3-\left (48 b^3\right ) \int \frac {1}{\sqrt {1-\frac {2}{d x^2}}} \, dx\\ &=24 a b^2 x-48 b^3 \sqrt {1-\frac {2}{d x^2}} x+24 b^3 x \cosh ^{-1}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^3\\ \end {align*}
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Mathematica [A] time = 0.13, size = 171, normalized size = 1.55 \[ \frac {a d x^2 \left (a^2+24 b^2\right )-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2} \sqrt {d x^2-2}+3 b \cosh ^{-1}\left (d x^2-1\right ) \left (a^2 d x^2-4 a b \sqrt {d x^2} \sqrt {d x^2-2}+8 b^2 d x^2\right )+3 b^2 \cosh ^{-1}\left (d x^2-1\right )^2 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2-2}\right )+b^3 d x^2 \cosh ^{-1}\left (d x^2-1\right )^3}{d x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 210, normalized size = 1.91 \[ \frac {b^{3} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \, {\left (a b^{2} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 3 \, {\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \]
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 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 3 \, a b^{2} x \operatorname {arcosh}\left (d x^{2} - 1\right )^{2} + 12 \, a b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d x^{2}} - 1\right )}{\sqrt {d x^{2} - 2} d^{2}}\right )} + 3 \, {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} a^{2} b + {\left (x \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right )^{3} - \int \frac {6 \, {\left (d^{2} x^{4} - 2 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - \sqrt {d} x\right )} \sqrt {d x^{2} - 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right )^{2}}{d^{2} x^{4} - 3 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - 2 \, \sqrt {d} x\right )} \sqrt {d x^{2} - 2} + 2}\,{d x}\right )} b^{3} + a^{3} x \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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