Optimal. Leaf size=352 \[ x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}-\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{4 \sqrt{2} a}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{4 \sqrt{2} a}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.200465, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.3, Rules used = {6170, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}-\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{4 \sqrt{2} a}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{4 \sqrt{2} a}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6170
Rule 94
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{1}{4} \coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^2 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )}{4 a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{4 a}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}\\ &=\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{2 a}-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 \sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0473046, size = 56, normalized size = 0.16 \[ \frac{2 e^{\frac{1}{4} \coth ^{-1}(a x)} \left (\left (e^{2 \coth ^{-1}(a x)}-1\right ) \text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},e^{2 \coth ^{-1}(a x)}\right )+1\right )}{a \left (e^{2 \coth ^{-1}(a x)}-1\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [8]{{\frac{ax-1}{ax+1}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52903, size = 358, normalized size = 1.02 \begin{align*} -\frac{1}{8} \, a{\left (\frac{16 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{2}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{2}} + \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.81266, size = 1224, normalized size = 3.48 \begin{align*} \frac{4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{3}{4}} + a^{2} \sqrt{\frac{1}{a^{4}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right ) + 4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{3}{4}} + a^{2} \sqrt{\frac{1}{a^{4}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right ) + \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{3}{4}} + a^{2} \sqrt{\frac{1}{a^{4}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{4}}^{\frac{3}{4}} + a^{2} \sqrt{\frac{1}{a^{4}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 8 \,{\left (a x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}} - 4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right ) + 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right ) - 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{8 \, a} \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 \frac{1}{\sqrt [8]{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22475, size = 363, normalized size = 1.03 \begin{align*} -\frac{1}{8} \, a{\left (\frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{2}} + \frac{2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{2}} - \frac{\sqrt{2} \log \left (\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} + \frac{\sqrt{2} \log \left (-\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{2}} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{2}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{2}} + \frac{2 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1 \right |}\right )}{a^{2}} + \frac{16 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]