Optimal. Leaf size=68 \[ \frac {x^3}{3}+\frac {x^3}{1-e^{2 a} x^4}+\frac {3}{2} e^{-3 a/2} \text {ArcTan}\left (e^{a/2} x\right )-\frac {3}{2} e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5657, 474, 470,
304, 209, 212} \begin {gather*} \frac {3}{2} e^{-3 a/2} \text {ArcTan}\left (e^{a/2} x\right )+\frac {x^3}{1-e^{2 a} x^4}-\frac {3}{2} e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 470
Rule 474
Rule 5657
Rubi steps
\begin {align*} \int x^2 \coth ^2(a+2 \log (x)) \, dx &=\int x^2 \coth ^2(a+2 \log (x)) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.12, size = 154, normalized size = 2.26 \begin {gather*} \frac {e^{-4 a} \left (-9317-17825 e^{2 a} x^4-4787 e^{4 a} x^8+1481 e^{6 a} x^{12}+7 \left (1331+1976 e^{2 a} x^4-398 e^{4 a} x^8-632 e^{6 a} x^{12}+27 e^{8 a} x^{16}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 a} x^4\right )\right )}{2688 x^5}+\frac {16 e^{2 a} x^7 \left (1+e^{2 a} x^4\right )^2 \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {19}{4};e^{2 a} x^4\right )}{1155} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.59, size = 100, normalized size = 1.47
method | result | size |
risch | \(\frac {x^{3}}{3}-\frac {x^{3}}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {3 \ln \left (-\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}+\frac {3 \ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left ({\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 66, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (49) = 98\).
time = 0.36, size = 104, normalized size = 1.53 \begin {gather*} \frac {4 \, x^{7} e^{\left (4 \, a\right )} - 16 \, x^{3} e^{\left (2 \, a\right )} + 18 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 9 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )}{12 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \coth ^{2}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 72, normalized size = 1.06 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.24, size = 60, normalized size = 0.88 \begin {gather*} \frac {3\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}-\frac {x^3}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^3}{3}+\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________