Optimal. Leaf size=68 \[ -\frac {2^{1+\frac {n}{2}} c^3 (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 71}
\begin {gather*} -\frac {c^3 2^{\frac {n}{2}+1} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 6264
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int (1-a x)^{3-\frac {n}{2}} (1+a x)^{n/2} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} c^3 (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 65, normalized size = 0.96 \begin {gather*} \frac {2^{1+\frac {n}{2}} c^3 (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (-8+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-c x a +c \right )^{3}\, dx\]
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^{3} \left (\int 3 a x e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- 3 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{3} x^{3} 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\,c\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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