Optimal. Leaf size=133 \[ \frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\text {ArcSin}(a x)}{a^7 c^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6283, 833, 655,
222} \begin {gather*} -\frac {\text {ArcSin}(a x)}{a^7 c^3}+\frac {x^5 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}+\frac {x (8 a x+5)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {x^3 (6 a x+5)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 833
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^6 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {x^2 (15+24 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {15+48 a x}{\sqrt {1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^6 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\sin ^{-1}(a x)}{a^7 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 108, normalized size = 0.81 \begin {gather*} \frac {48-33 a x-87 a^2 x^2+52 a^3 x^3+38 a^4 x^4-15 a^5 x^5-15 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{15 a^7 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(416\) vs.
\(2(117)=234\).
time = 0.07, size = 417, normalized size = 3.14
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a^{7} \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{6} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 a^{9} \left (x +\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 a^{8} \left (x +\frac {1}{a}\right )}+\frac {493 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{240 a^{8} \left (x -\frac {1}{a}\right )}+\frac {23 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{60 a^{9} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{20 a^{10} \left (x -\frac {1}{a}\right )^{3}}}{c^{3}}\) | \(259\) |
default | \(-\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{7}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{6} \sqrt {a^{2}}}-\frac {9 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{8} \left (x +\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{8}}+\frac {39 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 a^{8} \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{9}}+\frac {\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{12 a \left (x -\frac {1}{a}\right )^{2}}-\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{12 \left (x -\frac {1}{a}\right )}}{a^{8}}}{c^{3}}\) | \(417\) |
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 = 213, normalized size = 1.60 \begin {gather*} \frac {48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} - 48}{15 \, {\left (a^{12} c^{3} x^{5} - a^{11} c^{3} x^{4} - 2 \, a^{10} c^{3} x^{3} + 2 \, a^{9} c^{3} x^{2} + a^{8} c^{3} x - a^{7} c^{3}\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 {x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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 = 0.92, size = 379, normalized size = 2.85 \begin {gather*} \frac {a^6\,\sqrt {1-a^2\,x^2}}{30\,\left (a^{15}\,c^3\,x^2-2\,a^{14}\,c^3\,x+a^{13}\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^9\,c^3\,x^2+2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^5\,c^3\,\sqrt {-a^2}+3\,a^7\,c^3\,x^2\,\sqrt {-a^2}-a^8\,c^3\,x^3\,\sqrt {-a^2}-3\,a^6\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {5\,\sqrt {1-a^2\,x^2}}{12\,\left (a^9\,c^3\,x^2-2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {25\,\sqrt {1-a^2\,x^2}}{48\,\left (a^5\,c^3\,\sqrt {-a^2}+a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {493\,\sqrt {1-a^2\,x^2}}{240\,\left (a^5\,c^3\,\sqrt {-a^2}-a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a^7\,c^3}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^6\,c^3\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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