Optimal. Leaf size=287 \[ -\frac{12 a b x \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{24 a b \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{6 b^2 \sqrt{x} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{3 b^2 \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{3/2}}{d} \]
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Rubi [A] time = 0.448625, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5437, 4190, 4182, 2531, 6609, 2282, 6589, 4184, 3716, 2190} \[ -\frac{12 a b x \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{24 a b \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{6 b^2 \sqrt{x} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{3 b^2 \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 5437
Rule 4190
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4184
Rule 3716
Rule 2190
Rubi steps
\begin{align*} \int x \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b \text{csch}(c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^3+2 a b x^3 \text{csch}(c+d x)+b^2 x^3 \text{csch}^2(c+d x)\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^2}{2}+(4 a b) \operatorname{Subst}\left (\int x^3 \text{csch}(c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^3 \text{csch}^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{(12 a b) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(12 a b) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int x^2 \coth (c+d x) \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{12 a b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{(24 a b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(24 a b) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x^2}{1-e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 a b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(24 a b) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}+\frac{(24 a b) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{2 b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 a b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b^2 \sqrt{x} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(24 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{(24 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^4}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{2 b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 a b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b^2 \sqrt{x} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{24 a b \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ &=-\frac{2 b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{3/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 a b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 a b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b^2 \sqrt{x} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{3 b^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{24 a b \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{24 a b \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}\\ \end{align*}
Mathematica [B] time = 11.3794, size = 616, normalized size = 2.15 \[ \frac{2 b \sinh ^2\left (c+d \sqrt{x}\right ) \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \left (6 \left (a d^2 x-b d \sqrt{x}\right ) \text{PolyLog}\left (2,-e^{-c-d \sqrt{x}}\right )-6 \left (a d^2 x+b d \sqrt{x}\right ) \text{PolyLog}\left (2,e^{-c-d \sqrt{x}}\right )+12 a d \sqrt{x} \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right )-12 a d \sqrt{x} \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right )+12 a \text{PolyLog}\left (4,-e^{-c-d \sqrt{x}}\right )-12 a \text{PolyLog}\left (4,e^{-c-d \sqrt{x}}\right )-6 b \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right )-6 b \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right )+2 a d^3 x^{3/2} \log \left (1-e^{-c-d \sqrt{x}}\right )-2 a d^3 x^{3/2} \log \left (e^{-c-d \sqrt{x}}+1\right )-\frac{2 b d^3 x^{3/2}}{e^{2 c}-1}+3 b d^2 x \log \left (1-e^{-c-d \sqrt{x}}\right )+3 b d^2 x \log \left (e^{-c-d \sqrt{x}}+1\right )\right )}{d^4 \left (a \sinh \left (c+d \sqrt{x}\right )+b\right )^2}+\frac{a^2 x^2 \sinh ^2\left (c+d \sqrt{x}\right ) \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2}{2 \left (a \sinh \left (c+d \sqrt{x}\right )+b\right )^2}+\frac{b^2 x^{3/2} \text{csch}\left (\frac{c}{2}\right ) \sinh \left (\frac{d \sqrt{x}}{2}\right ) \sinh ^2\left (c+d \sqrt{x}\right ) \text{csch}\left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2}{d \left (a \sinh \left (c+d \sqrt{x}\right )+b\right )^2}-\frac{b^2 x^{3/2} \text{sech}\left (\frac{c}{2}\right ) \sinh \left (\frac{d \sqrt{x}}{2}\right ) \sinh ^2\left (c+d \sqrt{x}\right ) \text{sech}\left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2}{d \left (a \sinh \left (c+d \sqrt{x}\right )+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int x \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 [A] time = 1.92349, size = 463, normalized size = 1.61 \begin{align*} \frac{1}{2} \, a^{2} x^{2} - \frac{4 \, b^{2} x^{\frac{3}{2}}}{d e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} - d} - \frac{4 \,{\left (d^{3} x^{\frac{3}{2}} \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 3 \, d^{2} x{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) - 6 \, d \sqrt{x}{\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (d \sqrt{x} + c\right )})\right )} a b}{d^{4}} + \frac{4 \,{\left (d^{3} x^{\frac{3}{2}} \log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 3 \, d^{2} x{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) - 6 \, d \sqrt{x}{\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )}) + 6 \,{\rm Li}_{4}(e^{\left (d \sqrt{x} + c\right )})\right )} a b}{d^{4}} + \frac{6 \,{\left (d^{2} x \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 2 \, d \sqrt{x}{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )})\right )} b^{2}}{d^{4}} + \frac{6 \,{\left (d^{2} x \log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 2 \, d \sqrt{x}{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )})\right )} b^{2}}{d^{4}} - \frac{a b d^{4} x^{2} + 2 \, b^{2} d^{3} x^{\frac{3}{2}}}{d^{4}} + \frac{a b d^{4} x^{2} - 2 \, b^{2} d^{3} x^{\frac{3}{2}}}{d^{4}} \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 (b^{2} x \operatorname{csch}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b x \operatorname{csch}\left (d \sqrt{x} + c\right ) + a^{2} x, 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 x \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{\left (b \operatorname{csch}\left (d \sqrt{x} + c\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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