Optimal. Leaf size=142 \[ \frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {3 x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6171, 96, 94, 93, 212, 206, 203} \[ \frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {3 x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 6171
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} \coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/4}}{x^3 \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}} x^2+\frac {3 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/4}}{x^2 \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{4 a}\\ &=-\frac {3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}} x^2-\frac {9 \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}} x^2-\frac {9 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a^2}\\ &=-\frac {3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}} x^2+\frac {9 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ &=-\frac {3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{4 a}+\frac {1}{2} \left (1-\frac {1}{a x}\right )^{7/4} \sqrt [4]{1+\frac {1}{a x}} x^2+\frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 70, normalized size = 0.49 \[ \frac {-\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (3 e^{2 \coth ^{-1}(a x)}-7\right )}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+9 \tan ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+9 \tanh ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 95, normalized size = 0.67 \[ \frac {2 \, {\left (2 \, a^{2} x^{2} - 3 \, a x - 5\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}} - 18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 141, normalized size = 0.99 \[ -\frac {1}{8} \, a {\left (\frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} - \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {9 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (\frac {7 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 152, normalized size = 1.07 \[ -\frac {1}{8} \, a {\left (\frac {4 \, {\left (7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} - 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} + \frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} - \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.19, size = 121, normalized size = 0.85 \[ \frac {9\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}-\frac {9\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}-\frac {\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{2}-\frac {7\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________