Optimal. Leaf size=54 \[ \frac {1}{5} x^5 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {2 x^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}}{15 a^2}+\frac {x^4}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6336, 30, 271, 264} \[ \frac {1}{5} x^5 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {2 x^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}}{15 a^2}+\frac {x^4}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 264
Rule 271
Rule 6336
Rubi steps
\begin {align*} \int e^{\text {csch}^{-1}(a x)} x^4 \, dx &=\frac {\int x^3 \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^2}} x^4 \, dx\\ &=\frac {x^4}{4 a}+\frac {1}{5} \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^5-\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x^2 \, dx}{5 a^2}\\ &=-\frac {2 \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^3}{15 a^2}+\frac {x^4}{4 a}+\frac {1}{5} \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^5\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 49, normalized size = 0.91 \[ \frac {x \sqrt {\frac {1}{a^2 x^2}+1} \left (3 a^4 x^4+a^2 x^2-2\right )}{15 a^4}+\frac {x^4}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 53, normalized size = 0.98 \[ \frac {15 \, a^{3} x^{4} + 4 \, {\left (3 \, a^{4} x^{5} + a^{2} x^{3} - 2 \, x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}}}{60 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 78, normalized size = 1.44 \[ -\frac {a^{2} x^{2} + 1}{2 \, a^{5}} + \frac {2 \, {\left | a \right |} \mathrm {sgn}\relax (x)}{15 \, a^{6}} + \frac {12 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 20 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) + 15 \, {\left (a^{2} x^{2} + 1\right )}^{2} a}{60 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 53, normalized size = 0.98 \[ \frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (a^{2} x^{2}+1\right ) \left (3 a^{2} x^{2}-2\right )}{15 a^{4}}+\frac {x^{4}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 50, normalized size = 0.93 \[ \frac {x^{4}}{4 \, a} + \frac {3 \, a^{2} x^{5} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 5 \, x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{15 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.18, size = 41, normalized size = 0.76 \[ \sqrt {\frac {1}{a^2\,x^2}+1}\,\left (\frac {x^5}{5}-\frac {2\,x}{15\,a^4}+\frac {x^3}{15\,a^2}\right )+\frac {x^4}{4\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.01, size = 63, normalized size = 1.17 \[ \frac {x^{4} \sqrt {a^{2} x^{2} + 1}}{5 a} + \frac {x^{4}}{4 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{15 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1}}{15 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________