Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2},x\right ) \]
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Rubi [A] time = 0.02, 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 = {} \[ \int \frac {1}{x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=\int \frac {1}{x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 120.97, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x \operatorname {sech}\left (d \sqrt {x} + c\right )^{2} + 2 \, a b x \operatorname {sech}\left (d \sqrt {x} + c\right ) + a^{2} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]
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 \[ -\frac {4 \, {\left (b^{3} \sqrt {x} e^{\left (d \sqrt {x} + c\right )} + a b^{2} \sqrt {x}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d \sqrt {x}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x e^{\left (d \sqrt {x}\right )} + {\left (a^{5} d - a^{3} b^{2} d\right )} x} + \frac {\log \relax (x)}{a^{2}} - \int \frac {2 \, {\left (a b^{2} \sqrt {x} + {\left (b^{3} \sqrt {x} e^{c} + {\left (2 \, a^{2} b d e^{c} - b^{3} d e^{c}\right )} x\right )} e^{\left (d \sqrt {x}\right )}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d \sqrt {x}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x^{2} e^{\left (d \sqrt {x}\right )} + {\left (a^{5} d - a^{3} b^{2} d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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 \[ \int \frac {1}{x \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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