Optimal. Leaf size=128 \[ \frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \]
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Rubi [A]
time = 0.12, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6283, 837, 849,
821, 272, 65, 214} \begin {gather*} -\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}+\frac {a x+1}{c x^3 \sqrt {1-a^2 x^2}}-\frac {3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 849
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1+a x}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {4 a^2+3 a^3 x}{x^4 \sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {\int \frac {-9 a^3-8 a^4 x}{x^3 \sqrt {1-a^2 x^2}} \, dx}{3 a^2 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}+\frac {\int \frac {16 a^4+9 a^5 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{6 a^2 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}+\frac {\left (3 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{2 c}\\ &=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 91, normalized size = 0.71 \begin {gather*} -\frac {2+3 a x+8 a^2 x^2-9 a^3 x^3-16 a^4 x^4+9 a^3 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{6 c x^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 140, normalized size = 1.09
method | result | size |
risch | \(\frac {10 a^{4} x^{4}+3 a^{3} x^{3}-8 a^{2} x^{2}-3 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {a^{3} \left (-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}-3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2 c}\) | \(117\) |
default | \(-\frac {\frac {5 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{c}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 148, normalized size = 1.16 \begin {gather*} -\frac {\frac {3 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c} - \frac {3 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c} + \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} - 1\right )} a^{4} + 2 \, a^{4}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c - \sqrt {-a^{2} x^{2} + 1} c}}{4 \, a} + \frac {8 \, a^{4} x^{4} - 4 \, a^{2} x^{2} - 1}{3 \, \sqrt {a x + 1} \sqrt {-a x + 1} c x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 107, normalized size = 0.84 \begin {gather*} \frac {6 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 9 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a c x^{4} - c x^{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 {a}{- a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs.
\(2 (110) = 220\).
time = 0.41, size = 283, normalized size = 2.21 \begin {gather*} -\frac {{\left (a^{4} + \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {18 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {69 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {3 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c {\left | a \right |}} - \frac {\frac {21 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{x^{3}}}{24 \, a^{2} c^{3} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 140, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{3\,c\,x^3}-\frac {a\,\sqrt {1-a^2\,x^2}}{2\,c\,x^2}-\frac {5\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c\,x}-\frac {a^4\,\sqrt {1-a^2\,x^2}}{\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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