Optimal. Leaf size=209 \[ -\frac {n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac {(1-a x)^{-n/2} (1+a x)^{n/2} \left (6+8 n+n^2+n^3-a n \left (6+n^2\right ) x\right )}{6 a^5 c n}+\frac {2^{-1+\frac {n}{2}} n \left (8+n^2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^5 c (2-n)} \]
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Rubi [A]
time = 0.18, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6285, 102, 158,
148, 71} \begin {gather*} \frac {2^{\frac {n}{2}-1} n \left (n^2+8\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^5 c (2-n)}+\frac {(a x+1)^{n/2} \left (-a \left (n^2+6\right ) n x+n^3+n^2+8 n+6\right ) (1-a x)^{-n/2}}{6 a^5 c n}-\frac {n x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{6 a^3 c}-\frac {x^3 (a x+1)^{n/2} (1-a x)^{-n/2}}{3 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 148
Rule 158
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx &=\frac {\int x^4 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \, dx}{c}\\ &=-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}-\frac {\int x^2 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} (-3-a n x) \, dx}{3 a^2 c}\\ &=-\frac {n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac {\int x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (2 a n+a^2 \left (6+n^2\right ) x\right ) \, dx}{6 a^4 c}\\ &=-\frac {n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac {(1-a x)^{-n/2} (1+a x)^{n/2} \left (6+8 n+n^2+n^3-a n \left (6+n^2\right ) x\right )}{6 a^5 c n}-\frac {\left (n \left (8+n^2\right )\right ) \int (1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}} \, dx}{6 a^4 c}\\ &=-\frac {n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac {(1-a x)^{-n/2} (1+a x)^{n/2} \left (6+8 n+n^2+n^3-a n \left (6+n^2\right ) x\right )}{6 a^5 c n}+\frac {2^{-1+\frac {n}{2}} n \left (8+n^2\right ) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{3 a^5 c (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 173, normalized size = 0.83 \begin {gather*} \frac {e^{n \tanh ^{-1}(a x)} \left (6 (2+n)+n \left (e^{2 \tanh ^{-1}(a x)} n \left (2+n^2\right ) \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};-e^{2 \tanh ^{-1}(a x)}\right )-(2+n) \left (a \left (2+n^2\right ) x+(n+2 a x) \left (-1+a^2 x^2\right )+\left (2+n^2\right ) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};-e^{2 \tanh ^{-1}(a x)}\right )\right )\right )-24 e^{2 \tanh ^{-1}(a x)} n \, _2F_1\left (2,1+\frac {n}{2};2+\frac {n}{2};-e^{2 \tanh ^{-1}(a x)}\right )\right )}{6 a^5 c n (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{4}}{-a^{2} c \,x^{2}+c}\, dx\]
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*} - \frac {\int \frac {x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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 = -0.00 \begin {gather*} \int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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