Optimal. Leaf size=68 \[ -\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6275, 71}
\begin {gather*} -\frac {c 2^{\frac {n}{2}+2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-1,2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 6275
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^{1-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}} \, dx\\ &=-\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 65, normalized size = 0.96 \begin {gather*} \frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (-1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (-4+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )\, dx\]
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 \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c \left (\int a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \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 = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-a^2\,c\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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