Optimal. Leaf size=150 \[ \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{2 b d x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5888} \[ \frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{2 b d x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )} \]
Antiderivative was successfully verified.
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Rule 5888
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2} \, dx &=-\frac {\sqrt {d x^2} \sqrt {-2+d x^2}}{2 b d x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 141, normalized size = 0.94 \[ \frac {\frac {\sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}\left (d x^2-1\right )}{2 b}\right )\right )}{\sqrt {1-\frac {2}{d x^2}}}-\frac {b \sqrt {d x^2} \sqrt {d x^2-2}}{a+b \cosh ^{-1}\left (d x^2-1\right )}}{2 b^2 d x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} \operatorname {arcosh}\left (d x^{2} - 1\right )^{2} + 2 \, a b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{\left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{2}}\, dx \]
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 \[ -\frac {d^{2} x^{4} - 3 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - 2 \, \sqrt {d} x\right )} \sqrt {d x^{2} - 2} + 2}{2 \, {\left (a b d^{2} x^{3} - 2 \, a b d x + {\left (b^{2} d^{2} x^{3} - 2 \, b^{2} d x + {\left (b^{2} d^{\frac {3}{2}} x^{2} - b^{2} \sqrt {d}\right )} \sqrt {d x^{2} - 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right ) + {\left (a b d^{\frac {3}{2}} x^{2} - a b \sqrt {d}\right )} \sqrt {d x^{2} - 2}\right )}} + \int \frac {d^{3} x^{6} - 3 \, d^{2} x^{4} + {\left (d^{2} x^{4} - d x^{2} + 2\right )} {\left (d x^{2} - 2\right )} + {\left (2 \, d^{\frac {5}{2}} x^{5} - 4 \, d^{\frac {3}{2}} x^{3} + \sqrt {d} x\right )} \sqrt {d x^{2} - 2} + 4}{2 \, {\left (a b d^{3} x^{6} - 4 \, a b d^{2} x^{4} + 4 \, a b d x^{2} + {\left (a b d^{2} x^{4} - 2 \, a b d x^{2} + a b\right )} {\left (d x^{2} - 2\right )} + {\left (b^{2} d^{3} x^{6} - 4 \, b^{2} d^{2} x^{4} + 4 \, b^{2} d x^{2} + {\left (b^{2} d^{2} x^{4} - 2 \, b^{2} d x^{2} + b^{2}\right )} {\left (d x^{2} - 2\right )} + 2 \, {\left (b^{2} d^{\frac {5}{2}} x^{5} - 3 \, b^{2} d^{\frac {3}{2}} x^{3} + 2 \, b^{2} \sqrt {d} x\right )} \sqrt {d x^{2} - 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right ) + 2 \, {\left (a b d^{\frac {5}{2}} x^{5} - 3 \, a b d^{\frac {3}{2}} x^{3} + 2 \, a b \sqrt {d} x\right )} \sqrt {d x^{2} - 2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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