Optimal. Leaf size=90 \[ \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x+\frac {2 c^4 \log (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6292,
6285, 90} \begin {gather*} \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 6285
Rule 6292
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\\ &=-\frac {c^4 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \frac {(1-a x)^3 (1+a x)^5}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \left (-a^8+\frac {1}{x^8}+\frac {2 a}{x^7}-\frac {2 a^2}{x^6}-\frac {6 a^3}{x^5}+\frac {6 a^5}{x^3}+\frac {2 a^6}{x^2}-\frac {2 a^7}{x}\right ) \, dx}{a^8}\\ &=\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x+\frac {2 c^4 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 90, normalized size = 1.00 \begin {gather*} \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x+\frac {2 c^4 \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.27, size = 64, normalized size = 0.71
method | result | size |
default | \(\frac {c^{4} \left (a^{8} x -\frac {3 a^{3}}{2 x^{4}}+\frac {1}{7 x^{7}}+\frac {2 a^{6}}{x}+\frac {3 a^{5}}{x^{2}}+2 a^{7} \ln \left (x \right )+\frac {a}{3 x^{6}}-\frac {2 a^{2}}{5 x^{5}}\right )}{a^{8}}\) | \(64\) |
risch | \(c^{4} x +\frac {2 a^{6} c^{4} x^{6}+3 a^{5} c^{4} x^{5}-\frac {3}{2} a^{3} c^{4} x^{3}-\frac {2}{5} a^{2} c^{4} x^{2}+\frac {1}{3} a \,c^{4} x +\frac {1}{7} c^{4}}{a^{8} x^{7}}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(81\) |
norman | \(\frac {a^{7} c^{4} x^{8}+\frac {c^{4}}{7 a}+\frac {c^{4} x}{3}-\frac {3 a^{2} c^{4} x^{3}}{2}+3 a^{4} c^{4} x^{5}+2 a^{5} c^{4} x^{6}-\frac {2 c^{4} a \,x^{2}}{5}}{a^{7} x^{7}}+\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(86\) |
meijerg | \(-\frac {c^{4} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}-\frac {6 c^{4} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {4 c^{4} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{4 x^{4} a^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}-\frac {c^{4} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{6 x^{6} a^{6}}-\frac {1}{5 x^{5} a^{5}}-\frac {1}{4 x^{4} a^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {c^{4} \ln \left (-a x +1\right )}{a}-\frac {4 c^{4} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {6 c^{4} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {4 c^{4} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 x^{4} a^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}+\frac {c^{4} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{7 x^{7} a^{7}}+\frac {1}{6 x^{6} a^{6}}+\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 x^{4} a^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}\) | \(457\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 81, normalized size = 0.90 \begin {gather*} c^{4} x + \frac {2 \, c^{4} \log \left (x\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 89, normalized size = 0.99 \begin {gather*} \frac {210 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.26, size = 88, normalized size = 0.98 \begin {gather*} \frac {a^{8} c^{4} x + 2 a^{7} c^{4} \log {\left (x \right )} + \frac {420 a^{6} c^{4} x^{6} + 630 a^{5} c^{4} x^{5} - 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 82, normalized size = 0.91 \begin {gather*} c^{4} x + \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.24, size = 65, normalized size = 0.72 \begin {gather*} \frac {c^4\,\left (\frac {a\,x}{3}-\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5+2\,a^6\,x^6+a^8\,x^8+2\,a^7\,x^7\,\ln \left (x\right )+\frac {1}{7}\right )}{a^8\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________