Optimal. Leaf size=2123 \[ \text{result too large to display} \]
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Rubi [A] time = 2.98188, antiderivative size = 2123, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5436, 4191, 3324, 3320, 2264, 2190, 2531, 6609, 2282, 6589, 5562} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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[Out]
Rule 5436
Rule 4191
Rule 3324
Rule 3320
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5562
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b \text{sech}(c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^5}{a^2}+\frac{b^2 x^5}{a^2 (b+a \cosh (c+d x))^2}-\frac{2 b x^5}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^3}{3 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^5}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^5}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{x^3}{3 a^2}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{x^5}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4 \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2}}-\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{b-\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}-\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{b+\sqrt{-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac{\left (40 b^2\right ) \operatorname{Subst}\left (\int x^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{\left (40 b^2\right ) \operatorname{Subst}\left (\int x^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{(20 b) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{(20 b) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (120 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (120 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{(80 b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{(80 b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{\left (10 b^3\right ) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{\left (10 b^3\right ) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{(240 b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{(240 b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{\left (40 b^3\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{\left (40 b^3\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_4\left (-\frac{a e^{c+d x}}{b-\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_4\left (-\frac{a e^{c+d x}}{b+\sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{(480 b) \operatorname{Subst}\left (\int x \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{(480 b) \operatorname{Subst}\left (\int x \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{\left (120 b^3\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{\left (120 b^3\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{\left (240 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{(480 b) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{(480 b) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int x \text{Li}_4\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}-\frac{(480 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{(480 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}-\frac{480 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{480 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{\left (240 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}\\ &=\frac{2 b^2 x^{5/2}}{a^2 \left (a^2-b^2\right ) d}+\frac{x^3}{3 a^2}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{10 b^2 x^2 \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{2 b^3 x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{4 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{40 b^2 x^{3/2} \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{10 b^3 x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{20 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{120 b^2 x \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac{40 b^3 x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{80 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}-\frac{240 b^2 \sqrt{x} \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac{120 b^3 x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac{240 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^4}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}-\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{240 b^2 \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^6}+\frac{240 b^3 \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac{480 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^5}+\frac{240 b^3 \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}-\frac{480 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}-\frac{240 b^3 \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^6}+\frac{480 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^6}+\frac{2 b^2 x^{5/2} \sinh \left (c+d \sqrt{x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}
Mathematica [A] time = 16.9345, size = 2245, normalized size = 1.06 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm sech} \left (c+d\sqrt{x}\right ) \right ) ^{-2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{b^{2} \operatorname{sech}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b \operatorname{sech}\left (d \sqrt{x} + c\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b \operatorname{sech}{\left (c + d \sqrt{x} \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b \operatorname{sech}\left (d \sqrt{x} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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