Integrand size = 12, antiderivative size = 117 \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=-\frac {x}{a^3}+\frac {(1-a x) (1+a x)^3}{4 a^4}+\frac {(1+a x)^2 \left (3-8 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4} \]
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Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6472, 1818, 1825, 1828, 12, 267} \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=\frac {(1-a x) (a x+1)^3}{4 a^4}+\frac {\sqrt {\frac {1-a x}{a x+1}} \left (4-3 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^3}{6 a^4}+\frac {\left (3-8 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^2}{6 a^4}-\frac {x}{a^3} \]
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Rule 12
Rule 267
Rule 1818
Rule 1825
Rule 1828
Rule 6472
Rubi steps \begin{align*} \text {integral}& = \int x^3 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2 \, dx \\ & = \frac {4 \text {Subst}\left (\int \frac {(-1+x) x (1+x)^5}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^4} \\ & = \frac {(1-a x) (1+a x)^3}{4 a^4}-\frac {\text {Subst}\left (\int \frac {24 x+32 x^2-32 x^3-32 x^4-8 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^4} \\ & = \frac {(1-a x) (1+a x)^3}{4 a^4}-\frac {\text {Subst}\left (\int \frac {x \left (24+32 x-32 x^2-32 x^3-8 x^4\right )}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^4} \\ & = \frac {(1-a x) (1+a x)^3}{4 a^4}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {\text {Subst}\left (\int \frac {-64-48 x+192 x^2+48 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{12 a^4} \\ & = \frac {(1-a x) (1+a x)^3}{4 a^4}+\frac {(1+a x)^2 \left (3-8 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}-\frac {\text {Subst}\left (\int -\frac {192 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{48 a^4} \\ & = \frac {(1-a x) (1+a x)^3}{4 a^4}+\frac {(1+a x)^2 \left (3-8 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {4 \text {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^4} \\ & = -\frac {x}{a^3}+\frac {(1-a x) (1+a x)^3}{4 a^4}+\frac {(1+a x)^2 \left (3-8 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.44 \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=\frac {x^2}{a^2}-\frac {x^4}{4}+\frac {2 (-1+a x) \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{3 a^4} \]
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Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {-\frac {1}{4} x^{4} a^{2}+\frac {1}{2} x^{2}}{a^{2}}+\frac {2 \sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right )}{3 a^{3}}+\frac {x^{2}}{2 a^{2}}\) | \(72\) |
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Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.53 \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=-\frac {3 \, a^{3} x^{4} - 12 \, a x^{2} - 8 \, {\left (a^{2} x^{3} - x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{12 \, a^{3}} \]
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\[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=\frac {\int 2 x\, dx + \int \left (- a^{2} x^{3}\right )\, dx + \int 2 a x^{2} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.36 \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=-\frac {1}{4} \, x^{4} + \frac {x^{2}}{a^{2}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, a^{4}} \]
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\[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=\int { x^{3} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \]
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Time = 4.68 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.54 \[ \int e^{2 \text {sech}^{-1}(a x)} x^3 \, dx=\frac {x^2}{a^2}-\frac {x^4}{4}-\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,x\,\sqrt {\frac {1}{a\,x}+1}}{3\,a^3}-\frac {2\,x^3\,\sqrt {\frac {1}{a\,x}+1}}{3\,a}\right ) \]
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