Optimal. Leaf size=1639 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.58666, antiderivative size = 1639, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5437, 4191, 3324, 3322, 2264, 2190, 2531, 6609, 2282, 6589, 5561} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5437
Rule 4191
Rule 3324
Rule 3322
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5561
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{(a+b \text{csch}(c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^4}{a^2}+\frac{b^2 x^4}{a^2 (b+a \sinh (c+d x))^2}-\frac{2 b x^4}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{5/2}}{5 a^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^4}{b+a \sinh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{(b+a \sinh (c+d x))^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=\frac{2 x^{5/2}}{5 a^2}-\frac{2 b^2 x^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{-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^4}{b+a \sinh (c+d x)} \, dx,x,\sqrt{x}\right )}{a^2 \left (a^2+b^2\right )}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,\sqrt{x}\right )}{a \left (a^2+b^2\right ) d}\\ &=-\frac{2 b^2 x^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}-\frac{2 b^2 x^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{-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^4}{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^4}{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 (8 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^3}{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 (8 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^3}{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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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{8 b^2 x^{3/2} \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^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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^4}{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^4}{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 (24 b^2\right ) \operatorname{Subst}\left (\int x^2 \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 (24 b^2\right ) \operatorname{Subst}\left (\int x^2 \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{(16 b) \operatorname{Subst}\left (\int x^3 \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{(16 b) \operatorname{Subst}\left (\int x^3 \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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{16 b x^{3/2} \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{24 b^2 x \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{16 b x^{3/2} \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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int x \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 (48 b^2\right ) \operatorname{Subst}\left (\int x \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{(48 b) \operatorname{Subst}\left (\int x^2 \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{(48 b) \operatorname{Subst}\left (\int x^2 \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 (8 b^3\right ) \operatorname{Subst}\left (\int x^3 \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 (8 b^3\right ) \operatorname{Subst}\left (\int x^3 \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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}-\frac{16 b x^{3/2} \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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}+\frac{16 b x^{3/2} \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{48 b^2 \sqrt{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{48 b x \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{48 b^2 \sqrt{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{48 b x \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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int \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 (48 b^2\right ) \operatorname{Subst}\left (\int \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{(96 b) \operatorname{Subst}\left (\int x \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{(96 b) \operatorname{Subst}\left (\int x \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 (24 b^3\right ) \operatorname{Subst}\left (\int x^2 \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 (24 b^3\right ) \operatorname{Subst}\left (\int x^2 \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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}-\frac{16 b x^{3/2} \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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}+\frac{16 b x^{3/2} \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}+\frac{48 b x \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}-\frac{48 b x \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{96 b \sqrt{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{96 b \sqrt{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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}+\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^5}+\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\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^5}+\frac{(96 b) \operatorname{Subst}\left (\int \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{(96 b) \operatorname{Subst}\left (\int \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 (48 b^3\right ) \operatorname{Subst}\left (\int x \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 (48 b^3\right ) \operatorname{Subst}\left (\int x \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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}-\frac{16 b x^{3/2} \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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}+\frac{16 b x^{3/2} \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}+\frac{48 b x \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}-\frac{48 b x \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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}-\frac{96 b \sqrt{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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}+\frac{96 b \sqrt{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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}+\frac{(96 b) \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 \sqrt{a^2+b^2} d^5}-\frac{(96 b) \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 \sqrt{a^2+b^2} d^5}-\frac{\left (48 b^3\right ) \operatorname{Subst}\left (\int \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 (48 b^3\right ) \operatorname{Subst}\left (\int \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^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}-\frac{16 b x^{3/2} \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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}+\frac{16 b x^{3/2} \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}+\frac{48 b x \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}-\frac{48 b x \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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}-\frac{96 b \sqrt{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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}+\frac{96 b \sqrt{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{96 b \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{96 b \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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}-\frac{\left (48 b^3\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 )^{3/2} d^5}+\frac{\left (48 b^3\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 )^{3/2} d^5}\\ &=-\frac{2 b^2 x^2}{a^2 \left (a^2+b^2\right ) d}+\frac{2 x^{5/2}}{5 a^2}+\frac{8 b^2 x^{3/2} \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^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^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{8 b^2 x^{3/2} \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^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^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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}-\frac{16 b x^{3/2} \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{24 b^2 x \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{8 b^3 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 )^{3/2} d^2}+\frac{16 b x^{3/2} \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}+\frac{48 b x \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{48 b^2 \sqrt{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{24 b^3 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 )^{3/2} d^3}-\frac{48 b x \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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}-\frac{96 b \sqrt{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{48 b^2 \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{48 b^3 \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 )^{3/2} d^4}+\frac{96 b \sqrt{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{48 b^3 \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{96 b \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{48 b^3 \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{96 b \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^2 \cosh \left (c+d \sqrt{x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}
Mathematica [A] time = 18.1888, size = 1761, normalized size = 1.07 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{3}{2}}} \left ( a+b{\rm csch} \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^{\frac{3}{2}}}{b^{2} \operatorname{csch}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b \operatorname{csch}\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^{\frac{3}{2}}}{\left (a + b \operatorname{csch}{\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^{\frac{3}{2}}}{{\left (b \operatorname{csch}\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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