Optimal. Leaf size=61 \[ -\frac {2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac {1}{2} (1-a x)\right )}{a c (2 p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6130, 23, 69} \[ -\frac {2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac {1}{2} (1-a x)\right )}{a c (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
Rubi steps
\begin {align*} \int e^{-2 p \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int (1-a x)^p (1+a x)^{-p} (c-a c x)^p \, dx\\ &=\left ((1-a x)^p (c-a c x)^{-p}\right ) \int (1+a x)^{-p} (c-a c x)^{2 p} \, dx\\ &=-\frac {2^{-p} (1-a x)^p (c-a c x)^{1+p} \, _2F_1\left (p,1+2 p;2 (1+p);\frac {1}{2} (1-a x)\right )}{a c (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.92 \[ -\frac {2^{-p} (1-a x)^{p+1} (c-a c x)^p \, _2F_1\left (p,2 p+1;2 p+2;\frac {1}{2}-\frac {a x}{2}\right )}{2 a p+a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}, x\right ) \]
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 \[ \int \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (-a c x +c \right )^{p} {\mathrm e}^{-2 p \arctanh \left (a x \right )}\, dx \]
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 \[ \int \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}\,{d x} \]
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 \[ \int {\mathrm {e}}^{-2\,p\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \]
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 \[ \int \left (- c \left (a x - 1\right )\right )^{p} e^{- 2 p \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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