Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {a \cosh (c+d x)+a}},x\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx &=\int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{a x^{2} \cosh \left (d x + c\right ) + a x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \left (d x + c\right ) + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a +a \cosh \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \left (d x + c\right ) + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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