Optimal. Leaf size=137 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}-\frac {a x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)} \]
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Rubi [A] time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6160, 6149, 808, 364} \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}-\frac {a x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 6149
Rule 6160
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int e^{-\tanh ^{-1}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} (1-a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx-\left (a \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{1-2 p} \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 112, normalized size = 0.82 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (a (2 p-1) x \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )-2 (p-1) \, _2F_1\left (\frac {1}{2}-p,\frac {1}{2}-p;\frac {3}{2}-p;a^2 x^2\right )\right )}{2 (p-1) (2 p-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a x + 1}, x\right ) \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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