Optimal. Leaf size=109 \[ \frac{23 a}{16 c^3 (1-a x)}-\frac{a}{16 c^3 (a x+1)}+\frac{3 a}{8 c^3 (1-a x)^2}+\frac{a}{12 c^3 (1-a x)^3}+\frac{2 a \log (x)}{c^3}-\frac{9 a \log (1-a x)}{4 c^3}+\frac{a \log (a x+1)}{4 c^3}-\frac{1}{c^3 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135269, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 88} \[ \frac{23 a}{16 c^3 (1-a x)}-\frac{a}{16 c^3 (a x+1)}+\frac{3 a}{8 c^3 (1-a x)^2}+\frac{a}{12 c^3 (1-a x)^3}+\frac{2 a \log (x)}{c^3}-\frac{9 a \log (1-a x)}{4 c^3}+\frac{a \log (a x+1)}{4 c^3}-\frac{1}{c^3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{1}{x^2 (1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{x^2}+\frac{2 a}{x}+\frac{a^2}{4 (-1+a x)^4}-\frac{3 a^2}{4 (-1+a x)^3}+\frac{23 a^2}{16 (-1+a x)^2}-\frac{9 a^2}{4 (-1+a x)}+\frac{a^2}{16 (1+a x)^2}+\frac{a^2}{4 (1+a x)}\right ) \, dx}{c^3}\\ &=-\frac{1}{c^3 x}+\frac{a}{12 c^3 (1-a x)^3}+\frac{3 a}{8 c^3 (1-a x)^2}+\frac{23 a}{16 c^3 (1-a x)}-\frac{a}{16 c^3 (1+a x)}+\frac{2 a \log (x)}{c^3}-\frac{9 a \log (1-a x)}{4 c^3}+\frac{a \log (1+a x)}{4 c^3}\\ \end{align*}
Mathematica [A] time = 0.0831823, size = 78, normalized size = 0.72 \[ \frac{\frac{69 a}{1-a x}-\frac{3 a}{a x+1}+\frac{18 a}{(a x-1)^2}-\frac{4 a}{(a x-1)^3}+96 a \log (x)-108 a \log (1-a x)+12 a \log (a x+1)-\frac{48}{x}}{48 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 94, normalized size = 0.9 \begin{align*} -{\frac{1}{{c}^{3}x}}+2\,{\frac{a\ln \left ( x \right ) }{{c}^{3}}}-{\frac{a}{16\,{c}^{3} \left ( ax+1 \right ) }}+{\frac{a\ln \left ( ax+1 \right ) }{4\,{c}^{3}}}-{\frac{a}{12\,{c}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{3\,a}{8\,{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{23\,a}{16\,{c}^{3} \left ( ax-1 \right ) }}-{\frac{9\,a\ln \left ( ax-1 \right ) }{4\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.991831, size = 140, normalized size = 1.28 \begin{align*} -\frac{15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \,{\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} + \frac{a \log \left (a x + 1\right )}{4 \, c^{3}} - \frac{9 \, a \log \left (a x - 1\right )}{4 \, c^{3}} + \frac{2 \, a \log \left (x\right )}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.66167, size = 377, normalized size = 3.46 \begin{align*} -\frac{30 \, a^{4} x^{4} - 48 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 46 \, a x - 3 \,{\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 27 \,{\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 24 \,{\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (x\right ) - 12}{12 \,{\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.12012, size = 104, normalized size = 0.95 \begin{align*} - \frac{15 a^{4} x^{4} - 24 a^{3} x^{3} - 7 a^{2} x^{2} + 23 a x - 6}{6 a^{4} c^{3} x^{5} - 12 a^{3} c^{3} x^{4} + 12 a c^{3} x^{2} - 6 c^{3} x} + \frac{2 a \log{\left (x \right )} - \frac{9 a \log{\left (x - \frac{1}{a} \right )}}{4} + \frac{a \log{\left (x + \frac{1}{a} \right )}}{4}}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15346, size = 119, normalized size = 1.09 \begin{align*} \frac{a \log \left ({\left | a x + 1 \right |}\right )}{4 \, c^{3}} - \frac{9 \, a \log \left ({\left | a x - 1 \right |}\right )}{4 \, c^{3}} + \frac{2 \, a \log \left ({\left | x \right |}\right )}{c^{3}} - \frac{15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} c^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]