Optimal. Leaf size=376 \[ \frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)} \]
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Rubi [A]
time = 0.27, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6285, 102,
165, 91, 80, 71, 92, 47, 37} \begin {gather*} -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 \left (4-n^2\right )}-\frac {(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^5 c^2 n \left (4-n^2\right )}+\frac {(1-n) (n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^5 c^2 (2-n)}+\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{a^5 c^2 (2-n)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{-n/2}}{a^5 c^2}+\frac {(n+3) x (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^4 c^2}-\frac {x^3 (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 71
Rule 80
Rule 91
Rule 92
Rule 102
Rule 165
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int x^4 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{c^2}\\ &=-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {\int x^2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} (-3-a n x) \, dx}{a^2 c^2}\\ &=-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {n \int x^2 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{a^2 c^2}+\frac {(3+n) \int x^2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \, dx}{a^2 c^2}\\ &=\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}+\frac {\int (1-a x)^{-n/2} (1+a x)^{-2+\frac {n}{2}} \left (-a (1-n)+a^2 n x\right ) \, dx}{a^5 c^2}+\frac {(3+n) \int (1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} (-1-a n x) \, dx}{a^4 c^2}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}+\frac {n \int (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2}-\frac {\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-2-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2 (2-n)}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac {\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, dx}{a^4 c^2 \left (4-n^2\right )}\\ &=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 117, normalized size = 0.31 \begin {gather*} \frac {e^{n \tanh ^{-1}(a x)} \left (-6+2 a n x+6 a^2 x^2+n^2 \left (1-2 a^2 x^2\right )+4 e^{2 \tanh ^{-1}(a x)} (-2+n) n \left (-1+a^2 x^2\right ) \, _2F_1\left (2,1+\frac {n}{2};2+\frac {n}{2};-e^{2 \tanh ^{-1}(a x)}\right )\right )}{a^5 c^2 (-2+n) n (2+n) \left (-1+a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{4}}{\left (-a^{2} c \,x^{2}+c \right )^{2}}\, dx\]
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 [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^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 )}}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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