Optimal. Leaf size=89 \[ \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (d x)^m \, _2F_1\left (\frac {1}{2},\frac {m}{2};\frac {m+2}{2};c^2 x^2\right )}{c m}+\frac {(d x)^m \, _2F_1\left (1,\frac {m}{2};\frac {m+2}{2};c^2 x^2\right )}{c m} \]
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Rubi [A] time = 0.27, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 125
Rule 364
Rule 6341
Rule 6677
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*}
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Mathematica [F] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {e^{\text {sech}^{-1}(c x)} (d x)^m}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (d x\right )^{m} c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + \left (d x\right )^{m}}{c^{3} x^{3} - c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right ) \left (d x \right )^{m}}{-c^{2} x^{2}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -d^{m} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{m}}{c^{3} x^{3} - c x}\,{d x} - d^{m} \int \frac {x^{m}}{2 \, {\left (c x + 1\right )}}\,{d x} - d^{m} \int \frac {x^{m}}{2 \, {\left (c x - 1\right )}}\,{d x} + \frac {d^{m} x^{m}}{c m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}+\frac {1}{c\,x}\right )\,{\left (d\,x\right )}^m}{c^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\left (d x\right )^{m}}{c^{2} x^{3} - x}\, dx + \int \frac {c x \left (d x\right )^{m} \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{3} - x}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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