Optimal. Leaf size=339 \[ \frac {x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),2-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {6 a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (3-2 p),2-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac {2 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^p}{(1-p) (1-a x) (a x+1)}+\frac {a^4 x^5 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (5-2 p),2-p;\frac {1}{2} (7-2 p);a^2 x^2\right )}{5-2 p}+\frac {2 a^3 x^4 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (2-p,2-p;3-p;a^2 x^2\right )}{2-p} \]
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Rubi [A] time = 0.34, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6159, 6129, 127, 95, 125, 364} \[ \frac {x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),2-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {6 a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (3-2 p),2-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac {a^4 x^5 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (5-2 p),2-p;\frac {1}{2} (7-2 p);a^2 x^2\right )}{5-2 p}+\frac {2 a^3 x^4 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (2-p,2-p;3-p;a^2 x^2\right )}{2-p}+\frac {2 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^p}{(1-p) (1-a x) (a x+1)} \]
Antiderivative was successfully verified.
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Rule 95
Rule 125
Rule 127
Rule 364
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int e^{4 \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^{4 \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)^{-2+p} (1+a x)^{2+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 (4 a x^{1-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+6 a^2 x^{2-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+4 a^3 x^{3-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+a^4 x^{4-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+x^{-2 p} (1-a x)^{-2+p} (1+a x)^{-2+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)^{-2+p} (1+a x)^{-2+p} \, dx+\left (4 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)^{-2+p} (1+a x)^{-2+p} \, dx+\left (6 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)^{-2+p} (1+a x)^{-2+p} \, dx+\left (4 a^3 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{3-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx+\left (a^4 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{4-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx\\ &=\frac {2 a \left (c-\frac {c}{a^2 x^2}\right )^p x^2}{(1-p) (1-a x) (1+a x)}+\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 )^{-2+p} \, dx+\left (6 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 )^{-2+p} \, dx+\left (4 a^3 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{3-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx+\left (a^4 \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{4-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx\\ &=\frac {2 a \left (c-\frac {c}{a^2 x^2}\right )^p x^2}{(1-p) (1-a x) (1+a x)}+\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),2-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {6 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),2-p;\frac {1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^p x^5 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac {1}{2} (5-2 p),2-p;\frac {1}{2} (7-2 p);a^2 x^2\right )}{5-2 p}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^p x^4 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (2-p,2-p;3-p;a^2 x^2\right )}{2-p}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 217, normalized size = 0.64 \[ -\frac {x (1-a x)^{-p} \left (-\left (a^2 x^2-1\right )^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (-4 (a x+1) (a x-1)^p \left (1-a^2 x^2\right )^p F_1(1-2 p;1-p,-p;2-2 p;a x,-a x)+4 (a x-1)^p (a x+1)^{2 p} \left (1-a^2 x^2\right )^p \, _2F_1\left (1-2 p,2-p;2-2 p;\frac {2 a x}{a x+1}\right )+(a x+1) (1-a x)^p \left (a^2 x^2-1\right )^p \, _2F_1\left (\frac {1}{2}-p,-p;\frac {3}{2}-p;a^2 x^2\right )\right )}{(2 p-1) (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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