Optimal. Leaf size=52 \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{a c^3 (1-a x)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{a c^3 (1-a x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=\frac {\int \frac {(1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=\frac {\int \left (-\frac {4}{(-1+a x)^5}-\frac {4}{(-1+a x)^4}-\frac {1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=\frac {1}{a c^3 (1-a x)^4}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{2 a c^3 (1-a x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 31, normalized size = 0.60 \[ \frac {3 a^2 x^2+2 a x+1}{6 a c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 65, normalized size = 1.25 \[ \frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 42, normalized size = 0.81 \[ \frac {\frac {3}{{\left (a x - 1\right )}^{2} a} + \frac {8}{{\left (a x - 1\right )}^{3} a} + \frac {6}{{\left (a x - 1\right )}^{4} a}}{6 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 41, normalized size = 0.79 \[ \frac {\frac {1}{2 a \left (a x -1\right )^{2}}+\frac {4}{3 a \left (a x -1\right )^{3}}+\frac {1}{a \left (a x -1\right )^{4}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 65, normalized size = 1.25 \[ \frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.23, size = 29, normalized size = 0.56 \[ \frac {3\,a^2\,x^2+2\,a\,x+1}{6\,a\,c^3\,{\left (a\,x-1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.31, size = 70, normalized size = 1.35 \[ - \frac {- 3 a^{2} x^{2} - 2 a x - 1}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________