Optimal. Leaf size=89 \[ \frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (d x)^m \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2},\frac{m+2}{2},c^2 x^2\right )}{c m}+\frac{(d x)^m \text{Hypergeometric2F1}\left (1,\frac{m}{2},\frac{m+2}{2},c^2 x^2\right )}{c m} \]
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Rubi [A] time = 0.272119, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6341, 6677, 125, 364} \[ \frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (d x)^m \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{m+2}{2};c^2 x^2\right )}{c m}+\frac{(d x)^m \, _2F_1\left (1,\frac{m}{2};\frac{m+2}{2};c^2 x^2\right )}{c m} \]
Antiderivative was successfully verified.
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Rule 6341
Rule 6677
Rule 125
Rule 364
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx &=\frac{d \int \frac{(d x)^{-1+m} \sqrt{\frac{1}{1+c x}}}{\sqrt{1-c x}} \, dx}{c}+\frac{d \int \frac{(d x)^{-1+m}}{1-c^2 x^2} \, dx}{c}\\ &=\frac{(d x)^m \, _2F_1\left (1,\frac{m}{2};\frac{2+m}{2};c^2 x^2\right )}{c m}+\frac{\left (d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(d x)^{-1+m}}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=\frac{(d x)^m \, _2F_1\left (1,\frac{m}{2};\frac{2+m}{2};c^2 x^2\right )}{c m}+\frac{\left (d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(d x)^{-1+m}}{\sqrt{1-c^2 x^2}} \, dx}{c}\\ &=\frac{(d x)^m \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \, _2F_1\left (\frac{1}{2},\frac{m}{2};\frac{2+m}{2};c^2 x^2\right )}{c m}+\frac{(d x)^m \, _2F_1\left (1,\frac{m}{2};\frac{2+m}{2};c^2 x^2\right )}{c m}\\ \end{align*}
Mathematica [F] time = 0.634565, size = 0, normalized size = 0. \[ \int \frac{e^{\text{sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{-{c}^{2}{x}^{2}+1} \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -d^{m} \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} x^{m}}{c^{3} x^{3} - c x}\,{d x} - d^{m} \int \frac{x^{m}}{2 \,{\left (c x + 1\right )}}\,{d x} - d^{m} \int \frac{x^{m}}{2 \,{\left (c x - 1\right )}}\,{d x} + \frac{d^{m} x^{m}}{c m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (d x\right )^{m} c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + \left (d x\right )^{m}}{c^{3} x^{3} - c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\left (d x\right )^{m}}{c^{2} x^{3} - x}\, dx + \int \frac{c x \left (d x\right )^{m} \sqrt{-1 + \frac{1}{c x}} \sqrt{1 + \frac{1}{c x}}}{c^{2} x^{3} - x}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (d x\right )^{m}{\left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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