Optimal. Leaf size=73 \[ -\frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{2 c^3 (1-a x)^2}{a}-\frac{16 c^3 \log (a x+1)}{a}+8 c^3 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0645011, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{2 c^3 (1-a x)^2}{a}-\frac{16 c^3 \log (a x+1)}{a}+8 c^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=-\left (c^3 \int \frac{(1-a x)^4}{1+a x} \, dx\right )\\ &=-\left (c^3 \int \left (-8-4 (1-a x)-2 (1-a x)^2-(1-a x)^3+\frac{16}{1+a x}\right ) \, dx\right )\\ &=8 c^3 x-\frac{2 c^3 (1-a x)^2}{a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a}-\frac{16 c^3 \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0176466, size = 48, normalized size = 0.66 \[ -\frac{c^3 \left (3 a^4 x^4-20 a^3 x^3+66 a^2 x^2-180 a x+192 \log (a x+1)+35\right )}{12 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 53, normalized size = 0.7 \begin{align*} -{\frac{{c}^{3}{x}^{4}{a}^{3}}{4}}+{\frac{5\,{c}^{3}{x}^{3}{a}^{2}}{3}}-{\frac{11\,{c}^{3}{x}^{2}a}{2}}+15\,{c}^{3}x-16\,{\frac{{c}^{3}\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03387, size = 70, normalized size = 0.96 \begin{align*} -\frac{1}{4} \, a^{3} c^{3} x^{4} + \frac{5}{3} \, a^{2} c^{3} x^{3} - \frac{11}{2} \, a c^{3} x^{2} + 15 \, c^{3} x - \frac{16 \, c^{3} \log \left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45557, size = 130, normalized size = 1.78 \begin{align*} -\frac{3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log \left (a x + 1\right )}{12 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.601988, size = 56, normalized size = 0.77 \begin{align*} - \frac{a^{3} c^{3} x^{4}}{4} + \frac{5 a^{2} c^{3} x^{3}}{3} - \frac{11 a c^{3} x^{2}}{2} + 15 c^{3} x - \frac{16 c^{3} \log{\left (a x + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12859, size = 86, normalized size = 1.18 \begin{align*} -\frac{16 \, c^{3} \log \left ({\left | a x + 1 \right |}\right )}{a} - \frac{3 \, a^{7} c^{3} x^{4} - 20 \, a^{6} c^{3} x^{3} + 66 \, a^{5} c^{3} x^{2} - 180 \, a^{4} c^{3} x}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]