Optimal. Leaf size=37 \[ \frac {1}{2 a c^4 (1-a x)^4}-\frac {1}{3 a c^4 (1-a x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac {1}{2 a c^4 (1-a x)^4}-\frac {1}{3 a c^4 (1-a x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac {\int \frac {1+a x}{(1-a x)^5} \, dx}{c^4}\\ &=\frac {\int \left (-\frac {2}{(-1+a x)^5}-\frac {1}{(-1+a x)^4}\right ) \, dx}{c^4}\\ &=\frac {1}{2 a c^4 (1-a x)^4}-\frac {1}{3 a c^4 (1-a x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.62 \[ \frac {2 a x+1}{6 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 57, normalized size = 1.54 \[ \frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 21, normalized size = 0.57 \[ \frac {2 \, a x + 1}{6 \, {\left (a x - 1\right )}^{4} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 30, normalized size = 0.81 \[ \frac {\frac {1}{2 a \left (a x -1\right )^{4}}+\frac {1}{3 a \left (a x -1\right )^{3}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 57, normalized size = 1.54 \[ \frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 21, normalized size = 0.57 \[ \frac {2\,a\,x+1}{6\,a\,c^4\,{\left (a\,x-1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.29, size = 61, normalized size = 1.65 \[ - \frac {- 2 a x - 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________