Integrand size = 12, antiderivative size = 38 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {x^2}{a^2}+\frac {2 \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^3}{3 a}+\frac {x^4}{4} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6473, 6874, 270} \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {x^2}{a^2}+\frac {2 x^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}}{3 a}+\frac {x^4}{4} \]
[In]
[Out]
Rule 270
Rule 6473
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x^3 \, dx \\ & = \int \left (\frac {2 x}{a^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x^2}{a}+x^3\right ) \, dx \\ & = \frac {x^2}{a^2}+\frac {x^4}{4}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x^2 \, dx}{a} \\ & = \frac {x^2}{a^2}+\frac {2 \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^3}{3 a}+\frac {x^4}{4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {x^2}{a^2}+\frac {x^4}{4}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} \left (x+a^2 x^3\right )}{3 a^3} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55
method | result | size |
default | \(\frac {\left (a^{2} x^{2}+1\right )^{2}}{4 a^{4}}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (a^{2} x^{2}+1\right )}{3 a^{3}}+\frac {x^{2}}{2 a^{2}}\) | \(59\) |
trager | \(\frac {\frac {\left (a^{2} x^{3}+a^{2} x^{2}+a^{2} x +a^{2}+4 x +4\right ) \left (x -1\right )}{4}+\frac {2 \left (a^{2} x^{2}+1\right ) x \sqrt {-\frac {-a^{2} x^{2}-1}{a^{2} x^{2}}}}{3 a}}{a^{2}}\) | \(73\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {3 \, a^{3} x^{4} + 12 \, a x^{2} + 8 \, {\left (a^{2} x^{3} + x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}}}{12 \, a^{3}} \]
[In]
[Out]
Time = 1.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {x^{4}}{4} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{3 a^{2}} + \frac {x^{2}}{a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{3 a^{4}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {1}{4} \, x^{4} + \frac {2 \, x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{3 \, a} + \frac {x^{2}}{a^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\frac {a^{2} x^{2} + 1}{2 \, a^{4}} - \frac {2 \, {\left | a \right |} \mathrm {sgn}\left (x\right )}{3 \, a^{5}} + \frac {8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 3 \, {\left (a^{2} x^{2} + 1\right )}^{2} a^{3}}{12 \, a^{7}} \]
[In]
[Out]
Time = 5.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int e^{2 \text {csch}^{-1}(a x)} x^3 \, dx=\sqrt {\frac {1}{a^2\,x^2}+1}\,\left (\frac {2\,x}{3\,a^3}+\frac {2\,x^3}{3\,a}\right )+\frac {x^4}{4}+\frac {x^2}{a^2} \]
[In]
[Out]