Integrand size = 10, antiderivative size = 84 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {x^3}{12 a}+\frac {1}{4} e^{\text {sech}^{-1}(a x)} x^4-\frac {x \sqrt {1-a x}}{8 a^3 \sqrt {\frac {1}{1+a x}}}+\frac {\sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \arcsin (a x)}{8 a^4} \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 92, 41, 222} \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \arcsin (a x)}{8 a^4}-\frac {x \sqrt {1-a x}}{8 a^3 \sqrt {\frac {1}{a x+1}}}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a} \]
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Rule 30
Rule 41
Rule 92
Rule 222
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} e^{\text {sech}^{-1}(a x)} x^4+\frac {\int x^2 \, dx}{4 a}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 a} \\ & = \frac {x^3}{12 a}+\frac {1}{4} e^{\text {sech}^{-1}(a x)} x^4-\frac {x \sqrt {1-a x}}{8 a^3 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 a^3} \\ & = \frac {x^3}{12 a}+\frac {1}{4} e^{\text {sech}^{-1}(a x)} x^4-\frac {x \sqrt {1-a x}}{8 a^3 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3} \\ & = \frac {x^3}{12 a}+\frac {1}{4} e^{\text {sech}^{-1}(a x)} x^4-\frac {x \sqrt {1-a x}}{8 a^3 \sqrt {\frac {1}{1+a x}}}+\frac {\sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \arcsin (a x)}{8 a^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 a^3 x^3-3 a \sqrt {\frac {1-a x}{1+a x}} \left (x+a x^2-2 a^2 x^3-2 a^3 x^4\right )+3 i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{24 a^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (2 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-\sqrt {-a^{2} x^{2}+1}\, x \,\operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{8 \sqrt {-a^{2} x^{2}+1}\, a^{3}}+\frac {x^{3}}{3 a}\) | \(118\) |
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 \, a^{3} x^{3} + 3 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 3 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{24 \, a^{4}} \]
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\[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {\int x^{2}\, dx + \int a x^{3} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\int { x^{3} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} \,d x } \]
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Exception generated. \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\text {Exception raised: TypeError} \]
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Time = 16.44 (sec) , antiderivative size = 521, normalized size of antiderivative = 6.20 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{8\,a^4}-\frac {\frac {1{}\mathrm {i}}{1024\,a^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{128\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,11{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,7{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,239{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{8\,a^4}+\frac {x^3}{3\,a}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4} \]
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