Optimal. Leaf size=64 \[ -\frac {2 e^{\text {sech}^{-1}(a x)} x}{15 a^4}+\frac {x^2}{15 a^3}-\frac {e^{\text {sech}^{-1}(a x)} x^3}{15 a^2}+\frac {x^4}{20 a}+\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5 \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.30, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 102,
12, 75} \begin {gather*} -\frac {2 \sqrt {1-a x}}{15 a^5 \sqrt {\frac {1}{a x+1}}}-\frac {x^2 \sqrt {1-a x}}{15 a^3 \sqrt {\frac {1}{a x+1}}}+\frac {1}{5} x^5 e^{\text {sech}^{-1}(a x)}+\frac {x^4}{20 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 30
Rule 75
Rule 102
Rule 6470
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}(a x)} x^4 \, dx &=\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5+\frac {\int x^3 \, dx}{5 a}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x^3}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{5 a}\\ &=\frac {x^4}{20 a}+\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5-\frac {x^2 \sqrt {1-a x}}{15 a^3 \sqrt {\frac {1}{1+a x}}}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{15 a^3}\\ &=\frac {x^4}{20 a}+\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5-\frac {x^2 \sqrt {1-a x}}{15 a^3 \sqrt {\frac {1}{1+a x}}}+\frac {\left (2 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{15 a^3}\\ &=\frac {x^4}{20 a}+\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5-\frac {2 \sqrt {1-a x}}{15 a^5 \sqrt {\frac {1}{1+a x}}}-\frac {x^2 \sqrt {1-a x}}{15 a^3 \sqrt {\frac {1}{1+a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 65, normalized size = 1.02 \begin {gather*} \frac {15 a^4 x^4+4 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-2+2 a x-3 a^2 x^2+3 a^3 x^3\right )}{60 a^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 64, normalized size = 1.00
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (3 a^{2} x^{2}+2\right )}{15 a^{4}}+\frac {x^{4}}{4 a}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 47, normalized size = 0.73 \begin {gather*} \frac {x^{4}}{4 \, a} + \frac {{\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 65, normalized size = 1.02 \begin {gather*} \frac {15 \, a^{3} x^{4} + 4 \, {\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{60 \, a^{4}} \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 x^{3}\, dx + \int a x^{4} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.47, size = 75, normalized size = 1.17 \begin {gather*} \frac {x^4}{4\,a}-\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,x\,\sqrt {\frac {1}{a\,x}+1}}{15\,a^4}-\frac {x^5\,\sqrt {\frac {1}{a\,x}+1}}{5}+\frac {x^3\,\sqrt {\frac {1}{a\,x}+1}}{15\,a^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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