Optimal. Leaf size=138 \[ \frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6274, 669, 673,
198, 197} \begin {gather*} \frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}}+\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (a x+1) \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a c^4 (a x+1)^3 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 669
Rule 673
Rule 6274
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {(1-a x)^3}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=\frac {\int \frac {1}{(1+a x)^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^4}\\ &=-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{(1+a x)^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{3 c^4}\\ &=-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(1+a x) \left (1-a^2 x^2\right )^{5/2}} \, dx}{21 c^4}\\ &=-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{21 c^4}\\ &=\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{63 c^4}\\ &=\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 75, normalized size = 0.54 \begin {gather*} -\frac {19-6 a x-66 a^2 x^2-56 a^3 x^3+24 a^4 x^4+48 a^5 x^5+16 a^6 x^6}{63 a c^4 (1-a x)^{3/2} (1+a x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.07, size = 1697, normalized size = 12.30
method | result | size |
gosper | \(-\frac {16 x^{6} a^{6}+48 x^{5} a^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19}{63 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +1\right )^{3} c^{4} a}\) | \(74\) |
trager | \(-\frac {\left (16 x^{6} a^{6}+48 x^{5} a^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right ) \sqrt {-a^{2} x^{2}+1}}{63 c^{4} \left (a x +1\right )^{5} \left (a x -1\right )^{2} a}\) | \(81\) |
default | \(\text {Expression too large to display}\) | \(1697\) |
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 = 0.36, size = 195, normalized size = 1.41 \begin {gather*} -\frac {19 \, a^{7} x^{7} + 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} - 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 19 \, a^{2} x^{2} + 57 \, a x + {\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} + 19}{63 \, {\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right )}} \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 {1}{a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \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 = 1.27, size = 156, normalized size = 1.13 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {197\,x}{1008\,c^4}-\frac {155}{1008\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{252\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}}{36\,a\,c^4\,{\left (a\,x+1\right )}^5}-\frac {23\,\sqrt {1-a^2\,x^2}}{336\,a\,c^4\,{\left (a\,x+1\right )}^3}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{63\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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