Optimal. Leaf size=137 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {a x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)} \]
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Rubi [A] time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6160, 6148, 808, 364} \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {a x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 6148
Rule 6160
Rubi steps
\begin {align*} \int e^{\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} \left (1-a^2 x^2\right )^{-p}\right ) \int e^{\tanh ^{-1}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} (1+a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx+\left (a \left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{1-2 p} \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p;\frac {1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )}{2 (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 112, normalized size = 0.82 \[ -\frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 (p-1) \, _2F_1\left (\frac {1}{2}-p,\frac {1}{2}-p;\frac {3}{2}-p;a^2 x^2\right )+a (2 p-1) x \, _2F_1\left (\frac {1}{2}-p,1-p;2-p;a^2 x^2\right )\right )}{2 (p-1) (2 p-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.93, size = 178, normalized size = 1.30 \[ \frac {a c^{p} x^{2} \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ 2, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right )} + \frac {c^{p} x \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1, - p \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {e^{2 i \pi }}{a^{2} x^{2}}} \right )}}{\sqrt {\pi } \Gamma \left (p + 1\right )} + \frac {c^{p} x \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2}, 1 \\ \frac {3}{2}, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{\sqrt {\pi } \Gamma \left (p + 1\right )} - \frac {c^{p} {G_{3, 3}^{2, 2}\left (\begin {matrix} -1, p & 1 \\-1, 0 & - \frac {1}{2} \end {matrix} \middle | {\frac {e^{i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 a \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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