Optimal. Leaf size=128 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-\frac {2 c^5 \log (x)}{a}+c^5 (-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 128, 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} \[ -\frac {4 c^5}{a^3 x^2}+\frac {2 c^5}{3 a^4 x^3}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{5 a^6 x^5}-\frac {4 c^5}{3 a^7 x^6}-\frac {3 c^5}{7 a^8 x^7}+\frac {c^5}{4 a^9 x^8}+\frac {c^5}{9 a^{10} x^9}-\frac {3 c^5}{a^2 x}-\frac {2 c^5 \log (x)}{a}+c^5 (-x) \]
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 )^5 \, dx &=-\frac {c^5 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}}\\ &=-\frac {c^5 \int \frac {(1-a x)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}}\\ &=-\frac {c^5 \int \left (a^{10}+\frac {1}{x^{10}}+\frac {2 a}{x^9}-\frac {3 a^2}{x^8}-\frac {8 a^3}{x^7}+\frac {2 a^4}{x^6}+\frac {12 a^5}{x^5}+\frac {2 a^6}{x^4}-\frac {8 a^7}{x^3}-\frac {3 a^8}{x^2}+\frac {2 a^9}{x}\right ) \, dx}{a^{10}}\\ &=\frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-c^5 x-\frac {2 c^5 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 128, normalized size = 1.00 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{4 a^9 x^8}-\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}+\frac {2 c^5}{5 a^6 x^5}+\frac {3 c^5}{a^5 x^4}+\frac {2 c^5}{3 a^4 x^3}-\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}-\frac {2 c^5 \log (x)}{a}+c^5 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 122, normalized size = 0.95 \[ -\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \relax (x) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 116, normalized size = 0.91 \[ -c^{5} x - \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 117, normalized size = 0.91 \[ \frac {c^{5}}{9 a^{10} x^{9}}+\frac {c^{5}}{4 a^{9} x^{8}}-\frac {3 c^{5}}{7 a^{8} x^{7}}-\frac {4 c^{5}}{3 a^{7} x^{6}}+\frac {2 c^{5}}{5 a^{6} x^{5}}+\frac {3 c^{5}}{a^{5} x^{4}}+\frac {2 c^{5}}{3 a^{4} x^{3}}-\frac {4 c^{5}}{x^{2} a^{3}}-\frac {3 c^{5}}{a^{2} x}-c^{5} x -\frac {2 c^{5} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 115, normalized size = 0.90 \[ -c^{5} x - \frac {2 \, c^{5} \log \relax (x)}{a} - \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.89, size = 90, normalized size = 0.70 \[ -\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \relax (x)-\frac {1}{9}\right )}{a^{10}\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.82, size = 126, normalized size = 0.98 \[ \frac {- a^{10} c^{5} x - 2 a^{9} c^{5} \log {\relax (x )} - \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________