Integrand size = 16, antiderivative size = 280 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=-\frac {5 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x}+\frac {30 b^2 \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \sinh ^2\left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x} \]
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Time = 0.09 (sec) , antiderivative size = 280, 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 = {6001, 5999} \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {30 b^2 \sinh ^2\left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{d x}+x \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^{5/2}-\frac {5 b \left (d x^4+2 x^2\right ) \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2+2}} \]
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Rule 5999
Rule 6001
Rubi steps \begin{align*} \text {integral}& = -\frac {5 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}+\left (15 b^2\right ) \int \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \, dx \\ & = -\frac {5 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{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 ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{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 ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x}+\frac {30 b^2 \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \sinh ^2\left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x} \\ \end{align*}
Time = 2.03 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.11 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\frac {x \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right ) \left (-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \left (-5 a b \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+\left (a^2+15 b^2\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+b^2 \text {arccosh}\left (1+d x^2\right )^2 \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )-b \text {arccosh}\left (1+d x^2\right ) \left (5 b \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )-2 a \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )\right )\right )\right )}{2 \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}} \]
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\[\int {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}+1\right )\right )}^{\frac {5}{2}}d x\]
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Exception generated. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\text {Timed out} \]
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\[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2} \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^{5/2} \,d x \]
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