Optimal. Leaf size=125 \[ \frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^2}}{c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A]
time = 0.23, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 866,
1819, 1821, 821, 272, 65, 214} \begin {gather*} -\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {a \sqrt {1-a^2 x^2}}{c x^2}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}+\frac {2 a^3 (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 1821
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 (c-a c x)} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^4 (c-a c x)^2} \, dx\\ &=\frac {\int \frac {(c+a c x)^2}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\int \frac {-c^2-2 a c^2 x-2 a^2 c^2 x^2-2 a^3 c^2 x^3}{x^4 \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}+\frac {\int \frac {6 a c^2+8 a^2 c^2 x+6 a^3 c^2 x^2}{x^3 \sqrt {1-a^2 x^2}} \, dx}{3 c^3}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^2}}{c x^2}-\frac {\int \frac {-16 a^2 c^2-18 a^3 c^2 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{6 c^3}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^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}{c}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^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 )}{2 c}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^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 )}{c}\\ &=\frac {2 a^3 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3}-\frac {a \sqrt {1-a^2 x^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 )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 91, normalized size = 0.73 \begin {gather*} -\frac {1+3 a x+7 a^2 x^2-9 a^3 x^3-14 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 )}{3 c x^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 142, normalized size = 1.14
method | result | size |
risch | \(\frac {8 a^{4} x^{4}+3 a^{3} x^{3}-7 a^{2} x^{2}-3 a x -1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {a^{3} \left (-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c}\) | \(116\) |
default | \(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {8 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-2 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+2 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c}\) | \(142\) |
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 = 107, normalized size = 0.86 \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 (14 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\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 x}{a x^{5} \sqrt {- a^{2} x^{2} + 1} - x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x^{5} \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.
\(2 (111) = 222\).
time = 0.43, size = 283, normalized size = 2.26 \begin {gather*} -\frac {{\left (a^{4} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {27 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {129 \, {\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 )}{c {\left | a \right |}} - \frac {\frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac {6 \, {\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 = 0.06, size = 140, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{3\,c\,x^3}-\frac {a\,\sqrt {1-a^2\,x^2}}{c\,x^2}-\frac {8\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c\,x}-\frac {2\,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}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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