Optimal. Leaf size=37 \[ \frac{c^4 (1-a x)^5}{5 a}-\frac{c^4 (1-a x)^4}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0323223, antiderivative size = 37, normalized size of antiderivative = 1., 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} \[ \frac{c^4 (1-a x)^5}{5 a}-\frac{c^4 (1-a x)^4}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c^4 \int (1-a x)^3 (1+a x) \, dx\\ &=c^4 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\\ &=-\frac{c^4 (1-a x)^4}{2 a}+\frac{c^4 (1-a x)^5}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0155135, size = 32, normalized size = 0.86 \[ c^4 \left (-\frac{1}{5} a^4 x^5+\frac{a^3 x^4}{2}-a x^2+x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.025, size = 29, normalized size = 0.8 \begin{align*}{c}^{4} \left ( -{\frac{{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}-a{x}^{2}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.947139, size = 50, normalized size = 1.35 \begin{align*} -\frac{1}{5} \, a^{4} c^{4} x^{5} + \frac{1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54512, size = 76, normalized size = 2.05 \begin{align*} -\frac{1}{5} \, a^{4} c^{4} x^{5} + \frac{1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.091998, size = 36, normalized size = 0.97 \begin{align*} - \frac{a^{4} c^{4} x^{5}}{5} + \frac{a^{3} c^{4} x^{4}}{2} - a c^{4} x^{2} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20767, size = 50, normalized size = 1.35 \begin{align*} -\frac{1}{5} \, a^{4} c^{4} x^{5} + \frac{1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]