Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx &=\int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 67.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{2} \operatorname {sech}\left (d x^{2} + c\right )^{2} + 2 \, a b x^{2} \operatorname {sech}\left (d x^{2} + c\right ) + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left (a^{3} d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d x^{2}\right )} + a b^{2} + {\left (a^{3} d - a b^{2} d\right )} x^{2} + {\left (b^{3} e^{c} + 2 \, {\left (a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d x^{2}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{3} e^{\left (2 \, d x^{2}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x^{3} e^{\left (d x^{2}\right )} + {\left (a^{5} d - a^{3} b^{2} d\right )} x^{3}} - \int \frac {3 \, a b^{2} + {\left (3 \, b^{3} e^{c} + 2 \, {\left (2 \, a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d x^{2}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{4} e^{\left (2 \, d x^{2}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x^{4} e^{\left (d x^{2}\right )} + {\left (a^{5} d - a^{3} b^{2} d\right )} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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