Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{(d x)^m}{\left (a+b \text{sech}^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.0273039, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(d x)^m}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(d x)^m}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx &=\int \frac{(d x)^m}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.805672, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b{\rm arcsech} \left (cx\right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{2} d^{m} x^{3} - d^{m} x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} x^{m} +{\left (c^{2} d^{m} x^{3} - d^{m} x\right )} x^{m}}{{\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) -{\left (b^{2} \log \left (c\right ) + b^{2} \log \left (x\right ) - a b\right )} \sqrt{c x + 1} \sqrt{-c x + 1} + a b -{\left (b^{2} c^{2} x^{2} - \sqrt{c x + 1} \sqrt{-c x + 1} b^{2} - b^{2}\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right ) +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \left (x\right )} + \int \frac{{\left (c^{2} d^{m}{\left (m + 3\right )} x^{2} - d^{m}{\left (m + 1\right )}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} x^{m} +{\left (c^{4} d^{m}{\left (m + 2\right )} x^{4} - c^{2} d^{m}{\left (3 \, m + 5\right )} x^{2} + 2 \, d^{m}{\left (m + 1\right )}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} x^{m} +{\left (c^{4} d^{m}{\left (m + 1\right )} x^{4} - 2 \, c^{2} d^{m}{\left (m + 1\right )} x^{2} + d^{m}{\left (m + 1\right )}\right )} x^{m}}{{\left (b^{2} c^{4} \log \left (c\right ) - a b c^{4}\right )} x^{4} -{\left (b^{2} \log \left (c\right ) + b^{2} \log \left (x\right ) - a b\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} - 2 \,{\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - 2 \,{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) + a b +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \left (x\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1} - a b -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} -{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} - 2 \,{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} + b^{2}\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right ) +{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d x\right )^{m}}{b^{2} \operatorname{arsech}\left (c x\right )^{2} + 2 \, a b \operatorname{arsech}\left (c x\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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