Optimal. Leaf size=218 \[ \frac {x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),1-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}-\frac {a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p}+\frac {a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (3-2 p),1-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p} \]
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Rubi [A] time = 0.26, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6159, 6129, 127, 125, 364} \[ \frac {x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),1-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (3-2 p),1-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}-\frac {a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p} \]
Antiderivative was successfully verified.
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Rule 125
Rule 127
Rule 364
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int e^{-2 \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} (1-a x)^{-p} (1+a x)^{-p}\right ) \int e^{-2 \tanh ^{-1}(a x)} x^{-2 p} (1-a x)^p (1+a x)^p \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{1+p} (1+a x)^{-1+p} \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int \left (-2 a x^{1-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+a^2 x^{2-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+x^{-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}\right ) \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx-\left (2 a \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{1-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx+\left (a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx-\left (2 a \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{1-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx+\left (a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac {1}{2} (1-2 p),1-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^3 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac {1}{2} (3-2 p),1-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 142, normalized size = 0.65 \[ \frac {x (1-a x)^{-p} \left (-\left (a^2 x^2-1\right )^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left ((1-a x)^p \left (a^2 x^2-1\right )^p \, _2F_1\left (\frac {1}{2}-p,-p;\frac {3}{2}-p;a^2 x^2\right )-2 (a x-1)^p \left (1-a^2 x^2\right )^p F_1(1-2 p;-p,1-p;2-2 p;a x,-a x)\right )}{2 p-1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a x - 1\right )} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a x + 1}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.77, size = 695, normalized size = 3.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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