Optimal. Leaf size=115 \[ \frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {16 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}-\frac {32 \sqrt {d x}}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6591, 2455, 16, 321, 329, 212, 208, 205} \[ \frac {2 \sqrt {d x} \text {PolyLog}\left (2,a x^2\right )}{d}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {16 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}-\frac {32 \sqrt {d x}}{d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 205
Rule 208
Rule 212
Rule 321
Rule 329
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^2\right )}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+4 \int \frac {\log \left (1-a x^2\right )}{\sqrt {d x}} \, dx\\ &=\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+\frac {(16 a) \int \frac {x \sqrt {d x}}{1-a x^2} \, dx}{d}\\ &=\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+\frac {(16 a) \int \frac {(d x)^{3/2}}{1-a x^2} \, dx}{d^2}\\ &=-\frac {32 \sqrt {d x}}{d}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+16 \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )} \, dx\\ &=-\frac {32 \sqrt {d x}}{d}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+\frac {32 \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{d}\\ &=-\frac {32 \sqrt {d x}}{d}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}+16 \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )+16 \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )\\ &=-\frac {32 \sqrt {d x}}{d}+\frac {16 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{a} \sqrt {d}}+\frac {8 \sqrt {d x} \log \left (1-a x^2\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^2\right )}{d}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 57, normalized size = 0.50 \[ \frac {5 x \Gamma \left (\frac {5}{4}\right ) \left (16 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};a x^2\right )+\text {Li}_2\left (a x^2\right )+4 \log \left (1-a x^2\right )-16\right )}{2 \Gamma \left (\frac {9}{4}\right ) \sqrt {d x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 156, normalized size = 1.36 \[ -\frac {2 \, {\left (16 \, d \left (\frac {1}{a d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {d^{2} \sqrt {\frac {1}{a d^{2}}} + d x} a d \left (\frac {1}{a d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a d \left (\frac {1}{a d^{2}}\right )^{\frac {3}{4}}\right ) - 4 \, d \left (\frac {1}{a d^{2}}\right )^{\frac {1}{4}} \log \left (d \left (\frac {1}{a d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, d \left (\frac {1}{a d^{2}}\right )^{\frac {1}{4}} \log \left (-d \left (\frac {1}{a d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - \sqrt {d x} {\left ({\rm Li}_2\left (a x^{2}\right ) + 4 \, \log \left (-a x^{2} + 1\right ) - 16\right )}\right )}}{d} \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{2}\right )}{\sqrt {d x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 137, normalized size = 1.19 \[ \frac {2 \polylog \left (2, a \,x^{2}\right ) \sqrt {d x}}{d}+\frac {8 \sqrt {d x}\, \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{d}+\frac {16 \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{d}+\frac {8 \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{d}-\frac {32 \sqrt {d x}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 128, normalized size = 1.11 \[ -\frac {2 \, {\left (8 \, \sqrt {d x} {\left (\log \relax (d) + 2\right )} - \frac {8 \, d \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}} - \sqrt {d x} {\rm Li}_2\left (a x^{2}\right ) - 4 \, \sqrt {d x} \log \left (-a d^{2} x^{2} + d^{2}\right ) + \frac {4 \, d \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}}\right )}}{d} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x^2\right )}{\sqrt {d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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