Optimal. Leaf size=54 \[ \frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-p,p;1+p;\frac {1}{2} (1+a x)\right )}{a p} \]
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Rubi [A]
time = 0.04, antiderivative size = 54, 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 = {6277, 692, 71}
\begin {gather*} \frac {2^{p+1} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (a x+1)\right )}{a p} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 692
Rule 6277
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{-1+p} \, dx\\ &=\left (c (1-a x)^{-p} (c+a c x)^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{1+p} (c+a c x)^{-1+p} \, dx\\ &=\frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-p,p;1+p;\frac {1}{2} (1+a x)\right )}{a p}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 74, normalized size = 1.37 \begin {gather*} -\frac {2^{-1+p} (1-a x)^{2+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1-p,2+p;3+p;\frac {1}{2} (1-a x)\right )}{a (2+p)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.21, size = 651, normalized size = 12.06 \begin {gather*} - a \left (\begin {cases} \frac {0^{p} x}{a} + \frac {0^{p} \log {\left (\frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \log {\left (-1 + \frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \operatorname {acoth}{\left (\frac {1}{a x} \right )}}{a^{2}} - \frac {c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {0^{p} x}{a} + \frac {0^{p} \log {\left (\frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \log {\left (1 - \frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \operatorname {atanh}{\left (\frac {1}{a x} \right )}}{a^{2}} - \frac {c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {0^{p} \log {\left (a^{2} x^{2} - 1 \right )}}{2 a} + \frac {0^{p} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {a c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} + \frac {a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {0^{p} \log {\left (- a^{2} x^{2} + 1 \right )}}{2 a} + \frac {0^{p} \operatorname {atanh}{\left (a x \right )}}{a} + \frac {a c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} + \frac {a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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