Optimal. Leaf size=52 \[ \frac{c^3 (a x+1)^7}{7 a}-\frac{2 c^3 (a x+1)^6}{3 a}+\frac{4 c^3 (a x+1)^5}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0499648, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 43} \[ \frac{c^3 (a x+1)^7}{7 a}-\frac{2 c^3 (a x+1)^6}{3 a}+\frac{4 c^3 (a x+1)^5}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (4 (1+a x)^4-4 (1+a x)^5+(1+a x)^6\right ) \, dx\\ &=\frac{4 c^3 (1+a x)^5}{5 a}-\frac{2 c^3 (1+a x)^6}{3 a}+\frac{c^3 (1+a x)^7}{7 a}\\ \end{align*}
Mathematica [A] time = 0.0211414, size = 31, normalized size = 0.6 \[ \frac{c^3 (a x+1)^5 \left (15 a^2 x^2-40 a x+29\right )}{105 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.025, size = 52, normalized size = 1. \begin{align*}{c}^{3} \left ({\frac{{x}^{7}{a}^{6}}{7}}+{\frac{{x}^{6}{a}^{5}}{3}}-{\frac{{x}^{5}{a}^{4}}{5}}-{x}^{4}{a}^{3}-{\frac{{x}^{3}{a}^{2}}{3}}+a{x}^{2}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.963406, size = 93, normalized size = 1.79 \begin{align*} \frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} - \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68886, size = 142, normalized size = 2.73 \begin{align*} \frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} - \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.108122, size = 70, normalized size = 1.35 \begin{align*} \frac{a^{6} c^{3} x^{7}}{7} + \frac{a^{5} c^{3} x^{6}}{3} - \frac{a^{4} c^{3} x^{5}}{5} - a^{3} c^{3} x^{4} - \frac{a^{2} c^{3} x^{3}}{3} + a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12947, size = 93, normalized size = 1.79 \begin{align*} \frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} - \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]