Optimal. Leaf size=54 \[ \frac {2^{p+1} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (a x+1)\right )}{a p} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6142, 678, 69} \[ \frac {2^{p+1} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (a x+1)\right )}{a p} \]
Antiderivative was successfully verified.
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Rule 69
Rule 678
Rule 6142
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{-1+p} \, dx\\ &=\left (c (1-a x)^{-p} (c+a c x)^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{1+p} (c+a c x)^{-1+p} \, dx\\ &=\frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-p,p;1+p;\frac {1}{2} (1+a x)\right )}{a p}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 1.37 \[ -\frac {2^{p-1} (1-a x)^{p+2} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1-p,p+2;p+3;\frac {1}{2} (1-a x)\right )}{a (p+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a x - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 1}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.90, size = 651, normalized size = 12.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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