Optimal. Leaf size=97 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6263, 1653,
807, 673, 665} \begin {gather*} \frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rule 807
Rule 1653
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{(c-a c x)^4} \, dx &=c \int \frac {x^2 \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^4 (1-a x)^4}+\frac {\int \frac {\left (4 a^2 c^2-3 a^3 c^2 x\right ) \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx}{a^4 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^4 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx}{7 a^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{35 a^2 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.44 \begin {gather*} -\frac {(1+a x)^{3/2} \left (-2+10 a x-23 a^2 x^2\right )}{105 a^3 c^4 (1-a x)^{7/2}} \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. \(435\) vs.
\(2(85)=170\).
time = 0.77, size = 436, normalized size = 4.49
method | result | size |
gosper | \(-\frac {\left (23 a^{2} x^{2}-10 a x +2\right ) \left (a x +1\right )^{2}}{105 \left (a x -1\right )^{3} c^{4} a^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(49\) |
trager | \(\frac {\left (23 a^{3} x^{3}+13 a^{2} x^{2}-8 a x +2\right ) \sqrt {-a^{2} x^{2}+1}}{105 c^{4} a^{3} \left (a x -1\right )^{4}}\) | \(50\) |
default | \(\frac {\frac {\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{4}}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )^{3}}-2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{a^{5}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{6}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}}{c^{4}}\) | \(436\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (82) = 164\).
time = 0.47, size = 197, normalized size = 2.03 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{7} c^{4} x^{4} - 4 \, a^{6} c^{4} x^{3} + 6 \, a^{5} c^{4} x^{2} - 4 \, a^{4} c^{4} x + a^{3} c^{4}\right )}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{6} c^{4} x^{3} - 3 \, a^{5} c^{4} x^{2} + 3 \, a^{4} c^{4} x - a^{3} c^{4}\right )}} + \frac {82 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} c^{4} x^{2} - 2 \, a^{4} c^{4} x + a^{3} c^{4}\right )}} + \frac {23 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{4} x - a^{3} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 118, normalized size = 1.22 \begin {gather*} \frac {2 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 8 \, a x + {\left (23 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 8 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{105 \, {\left (a^{7} c^{4} x^{4} - 4 \, a^{6} c^{4} x^{3} + 6 \, a^{5} c^{4} x^{2} - 4 \, a^{4} c^{4} x + a^{3} 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 {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 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(-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 = 0.06, size = 347, normalized size = 3.58 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^7\,c^4\,x^4-4\,a^6\,c^4\,x^3+6\,a^5\,c^4\,x^2-4\,a^4\,c^4\,x+a^3\,c^4\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,c^4\,x^2-2\,a^4\,c^4\,x+a^3\,c^4\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {23\,\sqrt {1-a^2\,x^2}}{105\,\left (a\,c^4\,\sqrt {-a^2}-a^2\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}+\frac {29\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a\,c^4\,\sqrt {-a^2}+3\,a^3\,c^4\,x^2\,\sqrt {-a^2}-a^4\,c^4\,x^3\,\sqrt {-a^2}-3\,a^2\,c^4\,x\,\sqrt {-a^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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