Optimal. Leaf size=281 \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {30 b^2 \cosh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2}+\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2-2}} \]
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Rubi [A] time = 0.06, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5880, 5879} \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {30 b^2 \cosh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x}+\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 5879
Rule 5880
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2} \, dx &=\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}+\left (15 b^2\right ) \int \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )} \, dx\\ &=\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}+\frac {30 b^2 \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )} \cosh ^2\left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )}{d x}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\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 )}{d x}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\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 )}{d x}\\ \end {align*}
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Mathematica [A] time = 1.68, size = 277, normalized size = 0.99 \[ \frac {\cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (4 \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )} \left (\left (a^2+15 b^2\right ) \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )+b \cosh ^{-1}\left (d x^2-1\right ) \left (2 a \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )-5 b \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )\right )-5 a b \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )+b^2 \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \cosh ^{-1}\left (d x^2-1\right )^2\right )-15 \sqrt {2 \pi } b^{5/2} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )-15 \sqrt {2 \pi } b^{5/2} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )\right )}{2 d x} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{5/2} \,d x \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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