Optimal. Leaf size=102 \[ -\frac {e^{n \tanh ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {6 e^{n \tanh ^{-1}(a x)} (n-a x)}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6271, 6270}
\begin {gather*} -\frac {6 (n-a x) e^{n \tanh ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \tanh ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6270
Rule 6271
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac {e^{n \tanh ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c \left (9-n^2\right )}\\ &=-\frac {e^{n \tanh ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {6 e^{n \tanh ^{-1}(a x)} (n-a x)}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 123, normalized size = 1.21 \begin {gather*} -\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (-n^3-9 a x+3 a n^2 x+6 a^3 x^3+n \left (7-6 a^2 x^2\right )\right )}{a c^2 (-3+n) (-1+n) (1+n) (3+n) \sqrt {c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 84, normalized size = 0.82
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (6 a^{3} x^{3}-6 n \,x^{2} a^{2}+3 n^{2} x a -n^{3}-9 a x +7 n \right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(84\) |
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 [A]
time = 0.34, size = 166, normalized size = 1.63 \begin {gather*} -\frac {{\left (6 \, a^{3} x^{3} - 6 \, a^{2} n x^{2} - n^{3} + 3 \, {\left (a n^{2} - 3 \, a\right )} x + 7 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{4} - 10 \, a c^{3} n^{2} + {\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{4} + 9 \, a c^{3} - 2 \, {\left (a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3}\right )} x^{2}} \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*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \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 [B]
time = 1.21, size = 160, normalized size = 1.57 \begin {gather*} -\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {6\,x^3}{c^2\,\left (n^4-10\,n^2+9\right )}+\frac {7\,n-n^3}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {3\,x\,\left (n^2-3\right )}{a^2\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x^2}{a\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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