Optimal. Leaf size=114 \[ -\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x} \]
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Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6124, 835, 807, 266, 63, 208} \[ -\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 6124
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac {1+a x}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {1}{4} \int \frac {-4 a-3 a^2 x}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}+\frac {1}{12} \int \frac {9 a^2+8 a^3 x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {1}{24} \int \frac {-16 a^3-9 a^4 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{8} \left (3 a^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{16} \left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.66 \[ \frac {1}{24} \left (9 a^4 \log (x)-9 a^4 \log \left (\sqrt {1-a^2 x^2}+1\right )-\frac {\sqrt {1-a^2 x^2} \left (16 a^3 x^3+9 a^2 x^2+8 a x+6\right )}{x^4}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 69, normalized size = 0.61 \[ \frac {9 \, a^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 \, a x + 6\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 273, normalized size = 2.39 \[ \frac {{\left (3 \, a^{5} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} + \frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {3 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 0.88 \[ a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 107, normalized size = 0.94 \[ -\frac {3}{8} \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{3}}{3 \, x} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 98, normalized size = 0.86 \[ -\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {a\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {3\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.44, size = 258, normalized size = 2.26 \[ a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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