Optimal. Leaf size=110 \[ \frac {x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^2 (4+5 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(16+15 a x) \sqrt {1-a^2 x^2}}{6 a^6 c^2}+\frac {5 \text {ArcSin}(a x)}{2 a^6 c^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6283, 833, 794,
222} \begin {gather*} \frac {5 \text {ArcSin}(a x)}{2 a^6 c^2}+\frac {x^4 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {(15 a x+16) \sqrt {1-a^2 x^2}}{6 a^6 c^2}-\frac {x^2 (5 a x+4)}{3 a^4 c^2 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 794
Rule 833
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^5 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {x^3 (4+5 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^2 (4+5 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {x (8+15 a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^2 (4+5 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(16+15 a x) \sqrt {1-a^2 x^2}}{6 a^6 c^2}+\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^5 c^2}\\ &=\frac {x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^2 (4+5 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(16+15 a x) \sqrt {1-a^2 x^2}}{6 a^6 c^2}+\frac {5 \sin ^{-1}(a x)}{2 a^6 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 86, normalized size = 0.78 \begin {gather*} \frac {16-a x-23 a^2 x^2+3 a^3 x^3+3 a^4 x^4+15 (-1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{6 a^6 c^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. \(266\) vs.
\(2(96)=192\).
time = 0.08, size = 267, normalized size = 2.43
method | result | size |
risch | \(\frac {\left (a x +2\right ) \left (a^{2} x^{2}-1\right )}{2 a^{6} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{5} \sqrt {a^{2}}}+\frac {25 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{12 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{6 a^{8} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{7} \left (x +\frac {1}{a}\right )}}{c^{2}}\) | \(188\) |
default | \(\frac {\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{3}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{6}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{7} \left (x +\frac {1}{a}\right )}+\frac {9 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{4 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{2 a^{7}}}{c^{2}}\) | \(267\) |
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 = 152, normalized size = 1.38 \begin {gather*} -\frac {16 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 16 \, a x + 30 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - a x + 16\right )} \sqrt {-a^{2} x^{2} + 1} + 16}{6 \, {\left (a^{9} c^{2} x^{3} - a^{8} c^{2} x^{2} - a^{7} c^{2} x + a^{6} c^{2}\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^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 218, normalized size = 1.98 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^8\,c^2\,x^2-2\,a^7\,c^2\,x+a^6\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^4\,c^2\,\sqrt {-a^2}+a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {25\,\sqrt {1-a^2\,x^2}}{12\,\left (a^4\,c^2\,\sqrt {-a^2}-a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^6\,c^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^2}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^5\,c^2\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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