Optimal. Leaf size=239 \[ \frac {3 b \left (2 x^2-d x^4\right ) \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{d x}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{d x} \]
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Rubi [A]
time = 0.04, antiderivative size = 239, 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 = {6001, 6005}
\begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \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 {3 \sqrt {\frac {\pi }{2}} b^{3/2} \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}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}+\frac {3 b \left (2 x^2-d x^4\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{x \sqrt {d x^2} \sqrt {d x^2-2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6001
Rule 6005
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2} \, dx &=\frac {3 b \left (2 x^2-d x^4\right ) \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}+\left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}} \, dx\\ &=\frac {3 b \left (2 x^2-d x^4\right ) \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}+\frac {3 b^{3/2} \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 {3 b^{3/2} \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 = 0.32, size = 221, normalized size = 0.92 \begin {gather*} \frac {\cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \left (3 b^{3/2} \sqrt {2 \pi } \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 )-3 b^{3/2} \sqrt {2 \pi } \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 )+4 \sqrt {a+b \cosh ^{-1}\left (-1+d x^2\right )} \left (a \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )+b \cosh ^{-1}\left (-1+d x^2\right ) \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )-3 b \sinh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )\right )\right )}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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