Integrand size = 14, antiderivative size = 110 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=24 a b^2 x-48 b^3 \sqrt {1-\frac {2}{d x^2}} x+24 b^3 x \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \]
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Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6001, 6016, 12, 197} \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}+24 a b^2 x+24 b^3 x \text {arccosh}\left (d x^2-1\right )-48 b^3 x \sqrt {1-\frac {2}{d x^2}} \]
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Rule 12
Rule 197
Rule 6001
Rule 6016
Rubi steps \begin{align*} \text {integral}& = \frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+b \text {arccosh}\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 \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3+\left (24 b^3\right ) \int \text {arccosh}\left (-1+d x^2\right ) \, dx \\ & = 24 a b^2 x+24 b^3 x \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\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 \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\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 \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.55 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {a \left (a^2+24 b^2\right ) d x^2-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2} \sqrt {-2+d x^2}+3 b \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )+3 b^2 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^2+b^3 d x^2 \text {arccosh}\left (-1+d x^2\right )^3}{d x} \]
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\[\int {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{3}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (103) = 206\).
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.91 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {b^{3} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \, {\left (a b^{2} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 3 \, {\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \]
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\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \]
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\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{3} \,d x } \]
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Exception generated. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^3 \,d x \]
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