Optimal. Leaf size=63 \[ \frac {c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \, _2F_1\left (-p-2,p-1;p;\frac {1}{2} (1-a x)\right )}{a (1-p)} \]
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Rubi [A] time = 0.08, antiderivative size = 63, 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 = {6141, 678, 69} \[ \frac {c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \, _2F_1\left (-p-2,p-1;p;\frac {1}{2} (1-a x)\right )}{a (1-p)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 678
Rule 6141
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=c^2 \int (1+a x)^4 \left (c-a^2 c x^2\right )^{-2+p} \, dx\\ &=\left (c^2 (1+a x)^{1-p} (c-a c x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p}\right ) \int (1+a x)^{2+p} (c-a c x)^{-2+p} \, dx\\ &=\frac {2^{2+p} c (1+a x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p} \, _2F_1\left (-2-p,-1+p;p;\frac {1}{2} (1-a x)\right )}{a (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 1.14 \[ -\frac {2^{p+2} (1-a x)^{p-1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-2,p-1;p;\frac {1}{2} (1-a x)\right )}{a (p-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{4} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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