Integrand size = 10, antiderivative size = 64 \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=-\frac {2 e^{\text {sech}^{-1}(a x)} x}{15 a^4}+\frac {x^2}{15 a^3}-\frac {e^{\text {sech}^{-1}(a x)} x^3}{15 a^2}+\frac {x^4}{20 a}+\frac {1}{5} e^{\text {sech}^{-1}(a x)} x^5 \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 102, 12, 75} \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=-\frac {2 \sqrt {1-a x}}{15 a^5 \sqrt {\frac {1}{a x+1}}}-\frac {x^2 \sqrt {1-a x}}{15 a^3 \sqrt {\frac {1}{a x+1}}}+\frac {1}{5} x^5 e^{\text {sech}^{-1}(a x)}+\frac {x^4}{20 a} \]
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Rule 12
Rule 30
Rule 75
Rule 102
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\frac {15 a^4 x^4+4 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-2+2 a x-3 a^2 x^2+3 a^3 x^3\right )}{60 a^5} \]
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Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (3 a^{2} x^{2}+2\right )}{15 a^{4}}+\frac {x^{4}}{4 a}\) | \(64\) |
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Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\frac {15 \, a^{3} x^{4} + 4 \, {\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{60 \, a^{4}} \]
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\[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\frac {\int x^{3}\, dx + \int a x^{4} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \]
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\frac {x^{4}}{4 \, a} + \frac {{\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a^{5}} \]
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Exception generated. \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\text {Exception raised: TypeError} \]
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Time = 4.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17 \[ \int e^{\text {sech}^{-1}(a x)} x^4 \, dx=\frac {x^4}{4\,a}-\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,x\,\sqrt {\frac {1}{a\,x}+1}}{15\,a^4}-\frac {x^5\,\sqrt {\frac {1}{a\,x}+1}}{5}+\frac {x^3\,\sqrt {\frac {1}{a\,x}+1}}{15\,a^2}\right ) \]
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