Optimal. Leaf size=55 \[ \frac {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 659, 191} \[ \frac {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 659
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac {\int \frac {1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}}+\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 45, normalized size = 0.82 \[ \frac {2 a^2 x^2-2 a x-1}{3 a c^4 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 89, normalized size = 1.62 \[ \frac {a^{3} x^{3} - a^{2} x^{2} - a x - {\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{3 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{4} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 49, normalized size = 0.89 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}-2 a x -1\right )}{3 \left (a x -1\right )^{3} c^{4} a \left (a x +1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{4} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.83, size = 48, normalized size = 0.87 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^2\,x^2+2\,a\,x+1\right )}{3\,a\,c^4\,{\left (a\,x-1\right )}^2\,\left (a\,x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\right )\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________