Integrand size = 16, antiderivative size = 253 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x} \]
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Time = 0.05 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6010, 6005} \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{3 b^{5/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{3 b^{5/2} d x}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}+\frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}} \]
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Rule 6005
Rule 6010
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\int \frac {1}{\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}} \, dx}{3 b^2} \\ & = \frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=-\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {b} \left (\left (a+b \text {arccosh}\left (-1+d x^2\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )+b \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )\right )}{6 b^{5/2} d x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}} \]
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\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{5/2}} \,d x \]
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