Optimal. Leaf size=164 \[ -\frac{6 b x \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{12 b \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.173314, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {14, 5437, 4182, 2531, 6609, 2282, 6589} \[ -\frac{6 b x \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{12 b \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 5437
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right ) \, dx &=\int \left (a x+b x \text{csch}\left (c+d \sqrt{x}\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \text{csch}\left (c+d \sqrt{x}\right ) \, dx\\ &=\frac{a x^2}{2}+(2 b) \operatorname{Subst}\left (\int x^3 \text{csch}(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{6 b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{(12 b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(12 b) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{6 b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}+\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{6 b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(12 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{(12 b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^4}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{6 b x \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{6 b x \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \sqrt{x} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{12 b \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{12 b \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 2.53945, size = 181, normalized size = 1.1 \[ \frac{2 b \left (-3 d^2 x \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )+3 d^2 x \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )+6 d \sqrt{x} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )-6 d \sqrt{x} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )-6 \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )+6 \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )+d^3 x^{3/2} \log \left (1-e^{c+d \sqrt{x}}\right )-d^3 x^{3/2} \log \left (e^{c+d \sqrt{x}}+1\right )\right )}{d^4}+\frac{a x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm csch} \left (c+d\sqrt{x}\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.02263, size = 234, normalized size = 1.43 \begin{align*} \frac{1}{2} \, a x^{2} - \frac{2 \,{\left (\log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{3} + 3 \,{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt{x}\right )}\right ){\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (d \sqrt{x} + c\right )})\right )} b}{d^{4}} + \frac{2 \,{\left (\log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{3} + 3 \,{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) \log \left (e^{\left (d \sqrt{x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt{x}\right )}\right ){\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )}) + 6 \,{\rm Li}_{4}(e^{\left (d \sqrt{x} + c\right )})\right )} b}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x \operatorname{csch}\left (d \sqrt{x} + c\right ) + a x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d \sqrt{x} + c\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]