Optimal. Leaf size=197 \[ \frac {720 e^{n \tanh ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (64-20 n^2+n^4\right )}-\frac {e^{n \tanh ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \tanh ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )} \]
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Rubi [A]
time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6271, 6272}
\begin {gather*} -\frac {(n-6 a x) e^{n \tanh ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {360 (n-2 a x) e^{n \tanh ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac {30 (n-4 a x) e^{n \tanh ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac {720 e^{n \tanh ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6271
Rule 6272
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac {e^{n \tanh ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}+\frac {30 \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{c \left (36-n^2\right )}\\ &=-\frac {e^{n \tanh ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \tanh ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac {360 \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c^2 \left (576-52 n^2+n^4\right )}\\ &=-\frac {e^{n \tanh ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \tanh ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}+\frac {720 \int \frac {e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^3 \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}\\ &=\frac {720 e^{n \tanh ^{-1}(a x)}}{a c^4 n \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}-\frac {e^{n \tanh ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \tanh ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 170, normalized size = 0.86 \begin {gather*} -\frac {(1-a x)^{-3-\frac {n}{2}} (1+a x)^{-3+\frac {n}{2}} \left (-n^6+6 a n^5 x+n^4 \left (50-30 a^2 x^2\right )+120 a n^3 x \left (-2+a^2 x^2\right )-720 \left (-1+a^2 x^2\right )^3+48 a n x \left (33-40 a^2 x^2+15 a^4 x^4\right )-8 n^2 \left (68-105 a^2 x^2+45 a^4 x^4\right )\right )}{a c^4 (-6+n) (-4+n) (-2+n) n (2+n) (4+n) (6+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 167, normalized size = 0.85
method | result | size |
gosper | \(-\frac {\left (720 x^{6} a^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 n a x +544 n^{2}-720\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{3} c^{4} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\) | \(167\) |
risch | \(-\frac {\left (720 x^{6} a^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 n a x +544 n^{2}-720\right ) \left (-a x +1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{c^{4} \left (n^{2}-36\right ) \left (n^{2}-16\right ) \left (n^{2}-4\right ) a n \left (a^{2} x^{2}-1\right )^{3}}\) | \(183\) |
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 = 0.37, size = 310, normalized size = 1.57 \begin {gather*} \frac {{\left (720 \, a^{6} x^{6} - 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} - 120 \, {\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} - 6 \, {\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} - {\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 1.37, size = 301, normalized size = 1.53 \begin {gather*} \frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {n^6-50\,n^4+544\,n^2-720}{a^7\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {720\,x^5}{a^2\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {x^3\,\left (120\,n^2-1920\right )}{a^4\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {6\,x\,\left (n^4-40\,n^2+264\right )}{a^6\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^2\,\left (30\,n^4-840\,n^2+2160\right )}{a^5\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^4\,\left (360\,n^2-2160\right )}{a^3\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {1}{a^6}-x^6+\frac {3\,x^4}{a^2}-\frac {3\,x^2}{a^4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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