Optimal. Leaf size=110 \[ 24 a b^2 x+\frac{6 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2}{x \sqrt{d x^2} \sqrt{d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^3-48 b^3 x \sqrt{1-\frac{2}{d x^2}}+24 b^3 x \cosh ^{-1}\left (d x^2-1\right ) \]
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Rubi [A] time = 0.0456404, antiderivative size = 110, normalized size of antiderivative = 1., 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 5880
Rule 5901
Rule 12
Rule 191
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*}
Mathematica [A] time = 0.119618, 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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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}-1\right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10917, size = 441, normalized size = 4.01 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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