Optimal. Leaf size=115 \[ \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{1+a x}}} \]
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Rubi [A]
time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 105,
12, 97} \begin {gather*} \frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{a x+1}}}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{a x+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 97
Rule 105
Rule 6470
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}-\frac {\int \frac {1}{x^6} \, dx}{4 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 a}\\ &=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{20 a}\\ &=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}-\frac {1}{5} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {1}{15} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}-\frac {1}{15} \left (2 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{1+a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 60, normalized size = 0.52 \begin {gather*} \frac {-3+\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-3+3 a x-2 a^2 x^2+2 a^3 x^3\right )}{15 a x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 63, normalized size = 0.55
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (2 a^{2} x^{2}+3\right )}{15 x^{4}}-\frac {1}{5 a \,x^{5}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 51, normalized size = 0.44 \begin {gather*} \frac {{\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a x^{6}} - \frac {1}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 60, normalized size = 0.52 \begin {gather*} \frac {{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 3}{15 \, a x^{5}} \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 {1}{x^{6}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{5}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.56, size = 76, normalized size = 0.66 \begin {gather*} \frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{15}-\frac {\sqrt {\frac {1}{a\,x}+1}}{5}+\frac {2\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{15}\right )}{x^4}-\frac {1}{5\,a\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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