Optimal. Leaf size=103 \[ -\frac {16 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {16 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}+\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}} \]
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Rubi [A] time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6591, 2455, 16, 329, 298, 205, 208} \[ -\frac {2 \text {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}-\frac {16 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {16 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 205
Rule 208
Rule 298
Rule 329
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^2\right )}{(d x)^{3/2}} \, dx &=-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}-4 \int \frac {\log \left (1-a x^2\right )}{(d x)^{3/2}} \, dx\\ &=\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}+\frac {(16 a) \int \frac {x}{\sqrt {d x} \left (1-a x^2\right )} \, dx}{d}\\ &=\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}+\frac {(16 a) \int \frac {\sqrt {d x}}{1-a x^2} \, dx}{d^2}\\ &=\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}+\frac {(32 a) \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{d^3}\\ &=\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}+\frac {\left (16 \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{d}-\frac {\left (16 \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{d}\\ &=-\frac {16 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {16 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^2\right )}{d \sqrt {d x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 62, normalized size = 0.60 \[ \frac {x \Gamma \left (\frac {3}{4}\right ) \left (16 a x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};a x^2\right )-3 \text {Li}_2\left (a x^2\right )+12 \log \left (1-a x^2\right )\right )}{2 \Gamma \left (\frac {7}{4}\right ) (d x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.85, size = 170, normalized size = 1.65 \[ \frac {2 \, {\left (16 \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a d \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} - \sqrt {a d^{4} \sqrt {\frac {a}{d^{6}}} + a^{2} d x} d \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}}}{a}\right ) + 4 \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (512 \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) - 4 \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (-512 \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) - \sqrt {d x} {\left ({\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right )\right )}\right )}}{d^{2} x} \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{2}\right )}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 127, normalized size = 1.23 \[ -\frac {2 \polylog \left (2, a \,x^{2}\right )}{d \sqrt {d x}}+\frac {8 \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{d \sqrt {d x}}-\frac {16 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}+\frac {8 \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 123, normalized size = 1.19 \[ -\frac {2 \, {\left (4 \, a {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}}\right )} + \frac {{\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a d^{2} x^{2} + d^{2}\right ) + 8 \, \log \relax (d)}{\sqrt {d x}}\right )}}{d} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x^2\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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