Optimal. Leaf size=105 \[ -\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A]
time = 0.22, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6266, 6263,
866, 1819, 821, 272, 65, 214} \begin {gather*} -\frac {4 a^2 (4 a x+3)}{3 c \sqrt {1-a^2 x^2}}-\frac {8 a^2 (a x+1)}{3 c \left (1-a^2 x^2\right )^{3/2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \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 6263
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right ) x^3} \, dx &=-\frac {a \int \frac {e^{3 \tanh ^{-1}(a x)}}{x^2 (1-a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)^4} \, dx}{c}\\ &=-\frac {a \int \frac {(1+a x)^4}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}+\frac {a \int \frac {-3-12 a x-13 a^2 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}-\frac {a \int \frac {3+12 a x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{3 c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}-\frac {\left (4 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 \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 {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 92, normalized size = 0.88 \begin {gather*} \frac {a \left (-3+23 a x+7 a^2 x^2-19 a^3 x^3+12 a x (-1+a x) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{3 c x (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.12, size = 165, normalized size = 1.57
method | result | size |
default | \(\frac {a \left (-\frac {a^{2} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {1}{x \sqrt {-a^{2} x^{2}+1}}-4 a \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+8 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )\right )}{c}\) | \(165\) |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) a}{x \sqrt {-a^{2} x^{2}+1}\, c}-\frac {4 a^{2} \left (\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c}\) | \(181\) |
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 = 0.34, size = 120, normalized size = 1.14 \begin {gather*} -\frac {20 \, a^{4} x^{3} - 40 \, a^{3} x^{2} + 20 \, a^{2} x + 12 \, {\left (a^{4} x^{3} - 2 \, a^{3} x^{2} + a^{2} x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (19 \, a^{3} x^{2} - 26 \, a^{2} x + 3 \, a\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c x^{3} - 2 \, a c x^{2} + c x\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 {a \left (\int \frac {3 a x}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{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. 217 vs.
\(2 (93) = 186\).
time = 0.43, size = 217, normalized size = 2.07 \begin {gather*} \frac {4 \, a^{3} \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 {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{2 \, c x {\left | a \right |}} + \frac {{\left (3 \, a^{3} - \frac {89 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} + \frac {153 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} - \frac {99 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}}\right )} a^{2} x}{6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 138, normalized size = 1.31 \begin {gather*} \frac {a\,\sqrt {1-a^2\,x^2}}{c\,x}-\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )}+\frac {16\,a^3\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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