Optimal. Leaf size=97 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6263, 807, 673,
665} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rule 807
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{(c-a c x)^4} \, dx &=c \int \frac {x \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac {5 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx}{7 a}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{7 a c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.43 \begin {gather*} -\frac {(1+a x)^{3/2} \left (1-5 a x+a^2 x^2\right )}{21 a^2 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. \(394\) vs.
\(2(85)=170\).
time = 1.06, size = 395, normalized size = 4.07
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}-5 a x +1\right ) \left (a x +1\right )^{2}}{21 \left (a x -1\right )^{3} c^{4} a^{2} \sqrt {-a^{2} x^{2}+1}}\) | \(48\) |
trager | \(-\frac {\left (a^{3} x^{3}-4 a^{2} x^{2}-4 a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{21 c^{4} a^{2} \left (a x -1\right )^{4}}\) | \(49\) |
default | \(\frac {\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{3}}+\frac {\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{4}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{5}}}{c^{4}}\) | \(395\) |
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 = 0.48, size = 197, normalized size = 2.03 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{5} c^{4} x^{3} - 3 \, a^{4} c^{4} x^{2} + 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{21 \, {\left (a^{4} c^{4} x^{2} - 2 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{21 \, {\left (a^{3} c^{4} x - a^{2} 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 = 116, normalized size = 1.20 \begin {gather*} -\frac {a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + {\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{21 \, {\left (a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} 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}{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^{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}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 148, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (\frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} + \frac {28 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {21 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - 1\right )}}{21 \, a c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 295, normalized size = 3.04 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{21\,\left (c^4\,\sqrt {-a^2}-a\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {4\,a^2\,\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 {3\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (c^4\,\sqrt {-a^2}+3\,a^2\,c^4\,x^2\,\sqrt {-a^2}-a^3\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x\,\sqrt {-a^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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