Optimal. Leaf size=93 \[ \frac {11}{2} a^3 \csc ^{-1}(a x)-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.74, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6169, 1633, 1593, 12, 852, 1635, 1654, 780, 216} \[ -\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {11}{2} a^3 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 216
Rule 780
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1654
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^2}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (-a x^2-x^3\right )}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x) x^2 \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a^2 x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\operatorname {Subst}\left (\int \frac {x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2 \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-5-\frac {3 x}{a}\right ) \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {1}{2} \left (11 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {11}{2} a^3 \csc ^{-1}(a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 66, normalized size = 0.71 \[ \frac {1}{6} a \left (33 a^2 \sin ^{-1}\left (\frac {1}{a x}\right )+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (-52 a^3 x^3+19 a^2 x^2+7 a x+2\right )}{x^2 (a x-1)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 96, normalized size = 1.03 \[ -\frac {66 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (52 \, a^{4} x^{4} + 33 \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a x^{4} - x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 150, normalized size = 1.61 \[ -\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {12 \, a^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} + \frac {\frac {52 \, {\left (a x - 1\right )} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {21 \, {\left (a x - 1\right )}^{2} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 39 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 666, normalized size = 7.16 \[ \frac {-30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{6} a^{6}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+93 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{5} a^{5}+33 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, x^{5} a^{5}+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}-30 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-30 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}-51 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-96 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}-66 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}-12 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+60 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}+60 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+14 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+33 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+33 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-30 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-30 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+5 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}}{6 \sqrt {a^{2}}\, x^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 154, normalized size = 1.66 \[ -\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {75 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {88 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 12 \, a^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.24, size = 152, normalized size = 1.63 \[ -\frac {4\,a^3+\frac {88\,a^3\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {11\,a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {25\,a^3\,\left (a\,x-1\right )}{a\,x+1}}{\sqrt {\frac {a\,x-1}{a\,x+1}}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}-11\,a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________