Optimal. Leaf size=51 \[ -\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 51, 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, 45} \begin {gather*} -\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}+c^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6285
Rule 6292
Rule 6302
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx\\ &=\frac {c^2 \int \frac {e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \frac {(1+a x)^4}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \left (a^4+\frac {1}{x^4}+\frac {4 a}{x^3}+\frac {6 a^2}{x^2}+\frac {4 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 51, normalized size = 1.00 \begin {gather*} -\frac {c^2}{3 a^4 x^3}-\frac {2 c^2}{a^3 x^2}-\frac {6 c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 40, normalized size = 0.78
method | result | size |
default | \(\frac {c^{2} \left (a^{4} x -\frac {6 a^{2}}{x}-\frac {2 a}{x^{2}}+4 a^{3} \ln \left (x \right )-\frac {1}{3 x^{3}}\right )}{a^{4}}\) | \(40\) |
risch | \(c^{2} x +\frac {-6 a^{2} c^{2} x^{2}-2 a \,c^{2} x -\frac {1}{3} c^{2}}{a^{4} x^{3}}+\frac {4 c^{2} \ln \left (x \right )}{a}\) | \(48\) |
norman | \(\frac {-7 a^{3} c^{2} x^{4}+c^{2} a^{4} x^{5}+\frac {c^{2}}{3 a}+\frac {5 c^{2} x}{3}+4 a \,c^{2} x^{2}}{\left (a x -1\right ) a^{3} x^{3}}+\frac {4 c^{2} \ln \left (x \right )}{a}\) | \(71\) |
meijerg | \(-\frac {c^{2} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{2} x}{-a x +1}+\frac {c^{2} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {2 c^{2} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{2} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {2 c^{2} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {c^{2} \left (-\frac {5 a x}{-5 a x +5}+4 \ln \left (-a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}+\frac {3}{a x}\right )}{a}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 46, normalized size = 0.90 \begin {gather*} c^{2} x + \frac {4 \, c^{2} \log \left (x\right )}{a} - \frac {18 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.61, size = 56, normalized size = 1.10 \begin {gather*} \frac {3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} \log \left (x\right ) - 18 \, a^{2} c^{2} x^{2} - 6 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.11, size = 53, normalized size = 1.04 \begin {gather*} \frac {a^{4} c^{2} x + 4 a^{3} c^{2} \log {\left (x \right )} + \frac {- 18 a^{2} c^{2} x^{2} - 6 a c^{2} x - c^{2}}{3 x^{3}}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (49) = 98\).
time = 0.41, size = 112, normalized size = 2.20 \begin {gather*} -\frac {4 \, c^{2} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{2} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (3 \, c^{2} + \frac {34 \, c^{2}}{a x - 1} + \frac {66 \, c^{2}}{{\left (a x - 1\right )}^{2}} + \frac {36 \, c^{2}}{{\left (a x - 1\right )}^{3}}\right )} {\left (a x - 1\right )}}{3 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 43, normalized size = 0.84 \begin {gather*} -\frac {c^2\,\left (6\,a\,x+18\,a^2\,x^2-3\,a^4\,x^4-12\,a^3\,x^3\,\ln \left (x\right )+1\right )}{3\,a^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________