Optimal. Leaf size=15 \[ -\frac {c (a x+1)^3}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 37} \[ -\frac {c (a x+1)^3}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^4} \, dx &=c \int \frac {(1+a x)^2}{x^4} \, dx\\ &=-\frac {c (1+a x)^3}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {c (a x+1)^3}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 21, normalized size = 1.40 \[ -\frac {3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 21, normalized size = 1.40 \[ -\frac {3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 23, normalized size = 1.53 \[ c \left (-\frac {a^{2}}{x}-\frac {1}{3 x^{3}}-\frac {a}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 21, normalized size = 1.40 \[ -\frac {3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 21, normalized size = 1.40 \[ -\frac {c\,a^2\,x^2+c\,a\,x+\frac {c}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 24, normalized size = 1.60 \[ \frac {- 3 a^{2} c x^{2} - 3 a c x - c}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________