Optimal. Leaf size=68 \[ \frac {d \sqrt {d+e x^2}}{3 e^{3/2}}-\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6356, 272, 45}
\begin {gather*} -\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {d \sqrt {d+e x^2}}{3 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6356
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{3} \sqrt {e} \int \frac {x^3}{\sqrt {d+e x^2}} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \text {Subst}\left (\int \frac {x}{\sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \text {Subst}\left (\int \left (-\frac {d}{e \sqrt {d+e x}}+\frac {\sqrt {d+e x}}{e}\right ) \, dx,x,x^2\right )\\ &=\frac {d \sqrt {d+e x^2}}{3 e^{3/2}}-\frac {\left (d+e x^2\right )^{3/2}}{9 e^{3/2}}+\frac {1}{3} x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.82 \begin {gather*} \frac {1}{9} \left (\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2}}{e^{3/2}}+3 x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs.
\(2(50)=100\).
time = 0.01, size = 128, normalized size = 1.88
method | result | size |
default | \(\frac {x^{3} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{3}+\frac {e^{\frac {3}{2}} \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{3 d}-\frac {\sqrt {e}\, \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{3 d}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (50) = 100\).
time = 0.27, size = 102, normalized size = 1.50 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {artanh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}}\right ) - \frac {{\left (3 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} - 5 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d\right )} e^{\left (-\frac {3}{2}\right )}}{45 \, d} + \frac {{\left (3 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x^{2} e + d} d^{2}\right )} e^{\left (-\frac {3}{2}\right )}}{45 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (50) = 100\).
time = 0.37, size = 209, normalized size = 3.07 \begin {gather*} \frac {3 \, {\left (x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 3 \, x^{3} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right ) + 3 \, x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + x^{3} \sinh \left (\frac {1}{2}\right )^{3}\right )} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) - 2 \, {\left (x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 2 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + x^{2} \sinh \left (\frac {1}{2}\right )^{2} - 2 \, d\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{18 \, {\left (\cosh \left (\frac {1}{2}\right )^{3} + 3 \, \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right ) + 3 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + \sinh \left (\frac {1}{2}\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 65, normalized size = 0.96 \begin {gather*} \begin {cases} \frac {2 d \sqrt {d + e x^{2}}}{9 e^{\frac {3}{2}}} + \frac {x^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{3} - \frac {x^{2} \sqrt {d + e x^{2}}}{9 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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