Optimal. Leaf size=44 \[ -\frac {(c-a c x)^{p+2} \, _2F_1\left (1,p+2;p+3;\frac {1}{2} (1-a x)\right )}{2 a c^2 (p+2)} \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6130, 21, 68} \[ -\frac {(c-a c x)^{p+2} \, _2F_1\left (1,p+2;p+3;\frac {1}{2} (1-a x)\right )}{2 a c^2 (p+2)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 68
Rule 6130
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac {(1-a x) (c-a c x)^p}{1+a x} \, dx\\ &=\frac {\int \frac {(c-a c x)^{1+p}}{1+a x} \, dx}{c}\\ &=-\frac {(c-a c x)^{2+p} \, _2F_1\left (1,2+p;3+p;\frac {1}{2} (1-a x)\right )}{2 a c^2 (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.98 \[ \frac {(a x-1) (c-a c x)^p \left (\, _2F_1\left (1,p+1;p+2;\frac {1}{2} (1-a x)\right )-1\right )}{a (p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a x - 1\right )} {\left (-a c x + c\right )}^{p}}{a x + 1}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a c x +c \right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} {\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
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 \[ -\int \frac {\left (a^2\,x^2-1\right )\,{\left (c-a\,c\,x\right )}^p}{{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {\left (- a c x + c\right )^{p}}{a x + 1}\right )\, dx - \int \frac {a x \left (- a c x + c\right )^{p}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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