Optimal. Leaf size=83 \[ \frac {8}{15} a^5 \log (x)-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {7 a^3}{30 x^2}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {4}{15} a^5 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{5 x^5}-\frac {a}{20 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6012, 5916, 266, 44, 36, 29, 31} \[ \frac {7 a^3}{30 x^2}-\frac {4}{15} a^5 \log \left (1-a^2 x^2\right )+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}+\frac {8}{15} a^5 \log (x)-\frac {a^4 \tanh ^{-1}(a x)}{x}-\frac {a}{20 x^4}-\frac {\tanh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 5916
Rule 6012
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{x^6} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)}{x^6}-\frac {2 a^2 \tanh ^{-1}(a x)}{x^4}+\frac {a^4 \tanh ^{-1}(a x)}{x^2}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^6} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{5 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {1}{5} a \int \frac {1}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+a^5 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{5 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {1}{10} a \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} a^3 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}(a x)}{5 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {1}{10} a \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a^2}{x^2}+\frac {a^4}{x}-\frac {a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{3} a^3 \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^7 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a}{20 x^4}+\frac {7 a^3}{30 x^2}-\frac {\tanh ^{-1}(a x)}{5 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {8}{15} a^5 \log (x)-\frac {4}{15} a^5 \log \left (1-a^2 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 83, normalized size = 1.00 \[ \frac {8}{15} a^5 \log (x)-\frac {a^4 \tanh ^{-1}(a x)}{x}+\frac {7 a^3}{30 x^2}+\frac {2 a^2 \tanh ^{-1}(a x)}{3 x^3}-\frac {4}{15} a^5 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{5 x^5}-\frac {a}{20 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 81, normalized size = 0.98 \[ -\frac {16 \, a^{5} x^{5} \log \left (a^{2} x^{2} - 1\right ) - 32 \, a^{5} x^{5} \log \relax (x) - 14 \, a^{3} x^{3} + 3 \, a x + 2 \, {\left (15 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 265, normalized size = 3.19 \[ -\frac {4}{15} \, {\left (2 \, a^{4} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right ) - 2 \, a^{4} \log \left ({\left | -\frac {a x + 1}{a x - 1} - 1 \right |}\right ) + \frac {\frac {2 \, {\left (a x + 1\right )}^{3} a^{4}}{{\left (a x - 1\right )}^{3}} + \frac {7 \, {\left (a x + 1\right )}^{2} a^{4}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )} a^{4}}{a x - 1}}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{4}} - \frac {2 \, {\left (\frac {10 \, {\left (a x + 1\right )}^{2} a^{4}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{4}}{a x - 1} + a^{4}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{5}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 80, normalized size = 0.96 \[ -\frac {a^{4} \arctanh \left (a x \right )}{x}+\frac {2 a^{2} \arctanh \left (a x \right )}{3 x^{3}}-\frac {\arctanh \left (a x \right )}{5 x^{5}}-\frac {a}{20 x^{4}}+\frac {7 a^{3}}{30 x^{2}}+\frac {8 a^{5} \ln \left (a x \right )}{15}-\frac {4 a^{5} \ln \left (a x -1\right )}{15}-\frac {4 a^{5} \ln \left (a x +1\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 71, normalized size = 0.86 \[ -\frac {1}{60} \, {\left (16 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{4} \log \left (x^{2}\right ) - \frac {14 \, a^{2} x^{2} - 3}{x^{4}}\right )} a - \frac {{\left (15 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 3\right )} \operatorname {artanh}\left (a x\right )}{15 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.89, size = 70, normalized size = 0.84 \[ \frac {8\,a^5\,\ln \relax (x)}{15}-\frac {a}{20\,x^4}-\frac {\mathrm {atanh}\left (a\,x\right )}{5\,x^5}-\frac {4\,a^5\,\ln \left (a^2\,x^2-1\right )}{15}+\frac {7\,a^3}{30\,x^2}+\frac {2\,a^2\,\mathrm {atanh}\left (a\,x\right )}{3\,x^3}-\frac {a^4\,\mathrm {atanh}\left (a\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.93, size = 88, normalized size = 1.06 \[ \begin {cases} \frac {8 a^{5} \log {\relax (x )}}{15} - \frac {8 a^{5} \log {\left (x - \frac {1}{a} \right )}}{15} - \frac {8 a^{5} \operatorname {atanh}{\left (a x \right )}}{15} - \frac {a^{4} \operatorname {atanh}{\left (a x \right )}}{x} + \frac {7 a^{3}}{30 x^{2}} + \frac {2 a^{2} \operatorname {atanh}{\left (a x \right )}}{3 x^{3}} - \frac {a}{20 x^{4}} - \frac {\operatorname {atanh}{\left (a x \right )}}{5 x^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________