Optimal. Leaf size=147 \[ -\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {64 a^{7/4} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {64 a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {PolyLog}\left (2,a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6726, 2505, 16,
331, 335, 218, 214, 211} \begin {gather*} \frac {64 a^{7/4} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {64 a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}-\frac {128 a}{1029 d^3 (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 211
Rule 214
Rule 218
Rule 331
Rule 335
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{(d x)^{9/2}} \, dx &=-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {4}{7} \int \frac {\text {Li}_2\left (a x^2\right )}{(d x)^{9/2}} \, dx\\ &=-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}-\frac {16}{49} \int \frac {\log \left (1-a x^2\right )}{(d x)^{9/2}} \, dx\\ &=\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {(64 a) \int \frac {x}{(d x)^{7/2} \left (1-a x^2\right )} \, dx}{343 d}\\ &=\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {(64 a) \int \frac {1}{(d x)^{5/2} \left (1-a x^2\right )} \, dx}{343 d^2}\\ &=-\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {\left (64 a^2\right ) \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )} \, dx}{343 d^4}\\ &=-\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {\left (128 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{343 d^5}\\ &=-\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac {\left (64 a^2\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{343 d^4}+\frac {\left (64 a^2\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{343 d^4}\\ &=-\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {64 a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {64 a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.07, size = 84, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {d x} \Gamma \left (-\frac {3}{4}\right ) \left (-64 a x^2+192 a^2 x^4 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};a x^2\right )+48 \log \left (1-a x^2\right )-84 \text {PolyLog}\left (2,a x^2\right )-147 \text {PolyLog}\left (3,a x^2\right )\right )}{686 d^5 x^4 \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 142, normalized size = 0.97
method | result | size |
meijerg | \(-\frac {x^{\frac {9}{2}} \left (-a \right )^{\frac {7}{4}} \left (-\frac {256}{1029 x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{4}}}-\frac {64 \sqrt {x}\, a \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{343 \left (-a \right )^{\frac {3}{4}} \left (a \,x^{2}\right )^{\frac {1}{4}}}+\frac {64 \ln \left (-a \,x^{2}+1\right )}{343 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}-\frac {16 \polylog \left (2, a \,x^{2}\right )}{49 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}-\frac {4 \polylog \left (3, a \,x^{2}\right )}{7 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}\right )}{2 \left (d x \right )^{\frac {9}{2}}}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 168, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (\frac {48 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d} - \frac {a^{2} \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d}\right )}}{d^{2}} - \frac {64 \, a d^{2} x^{2} + 84 \, d^{2} {\rm Li}_2\left (a x^{2}\right ) - 48 \, d^{2} \log \left (-a d^{2} x^{2} + d^{2}\right ) + 96 \, d^{2} \log \left (d\right ) + 147 \, d^{2} {\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac {7}{2}} d^{2}}\right )}}{1029 \, 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. 223 vs.
\(2 (106) = 212\).
time = 0.41, size = 223, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (192 \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{2} d^{13} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {3}{4}} - \sqrt {d^{10} \sqrt {\frac {a^{7}}{d^{18}}} + a^{4} d x} d^{13} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {3}{4}}}{a^{7}}\right ) - 48 \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (32 \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) + 48 \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (-32 \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) + 4 \, {\left (16 \, a x^{2} + 21 \, {\rm Li}_2\left (a x^{2}\right ) - 12 \, \log \left (-a x^{2} + 1\right )\right )} \sqrt {d x} + 147 \, \sqrt {d x} {\rm polylog}\left (3, a x^{2}\right )\right )}}{1029 \, d^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\mathrm {polylog}\left (3,a\,x^2\right )}{{\left (d\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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