Optimal. Leaf size=78 \[ \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 (-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6157, 6150, 88} \[ \frac {2 c^3}{a^3 x^2}-\frac {c^3}{3 a^4 x^3}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{5 a^6 x^5}-\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1-a x)^4 (1+a x)^2}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \left (a^6+\frac {1}{x^6}-\frac {2 a}{x^5}-\frac {a^2}{x^4}+\frac {4 a^3}{x^3}-\frac {a^4}{x^2}-\frac {2 a^5}{x}\right ) \, dx}{a^6}\\ &=\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {2 c^3 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 78, normalized size = 1.00 \[ \frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}-\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}-\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 78, normalized size = 1.00 \[ -\frac {30 \, a^{6} c^{3} x^{6} - 60 \, a^{5} c^{3} x^{5} \log \relax (x) + 30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 136, normalized size = 1.74 \[ -\frac {2 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (30 \, c^{3} - \frac {71 \, c^{3}}{a x + 1} - \frac {65 \, c^{3}}{{\left (a x + 1\right )}^{2}} + \frac {310 \, c^{3}}{{\left (a x + 1\right )}^{3}} - \frac {270 \, c^{3}}{{\left (a x + 1\right )}^{4}} + \frac {60 \, c^{3}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}}{30 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 73, normalized size = 0.94 \[ \frac {c^{3}}{5 a^{6} x^{5}}-\frac {c^{3}}{2 a^{5} x^{4}}-\frac {c^{3}}{3 a^{4} x^{3}}+\frac {2 c^{3}}{x^{2} a^{3}}-\frac {c^{3}}{a^{2} x}-c^{3} x +\frac {2 c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 71, normalized size = 0.91 \[ -c^{3} x + \frac {2 \, c^{3} \log \relax (x)}{a} - \frac {30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.85, size = 57, normalized size = 0.73 \[ -\frac {c^3\,\left (\frac {a\,x}{2}+\frac {a^2\,x^2}{3}-2\,a^3\,x^3+a^4\,x^4+a^6\,x^6-2\,a^5\,x^5\,\ln \relax (x)-\frac {1}{5}\right )}{a^6\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.33, size = 76, normalized size = 0.97 \[ \frac {- a^{6} c^{3} x + 2 a^{5} c^{3} \log {\relax (x )} - \frac {30 a^{4} c^{3} x^{4} - 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} + 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________