Optimal. Leaf size=52 \[ -\frac {4 c^3 (1+a x)^5}{5 a}+\frac {2 c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6275
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x)^4 \, dx\right )\\ &=-\left (c^3 \int \left (4 (1+a x)^4-4 (1+a x)^5+(1+a x)^6\right ) \, dx\right )\\ &=-\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 = 0.02, size = 31, normalized size = 0.60 \begin {gather*} -\frac {c^3 (1+a x)^5 \left (29-40 a x+15 a^2 x^2\right )}{105 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 54, normalized size = 1.04
method | result | size |
gosper | \(-\frac {c^{3} x \left (15 a^{6} x^{6}+35 a^{5} x^{5}-21 a^{4} x^{4}-105 a^{3} x^{3}-35 a^{2} x^{2}+105 a x +105\right )}{105}\) | \(53\) |
default | \(c^{3} \left (-\frac {1}{7} a^{6} x^{7}-\frac {1}{3} a^{5} x^{6}+\frac {1}{5} a^{4} x^{5}+a^{3} x^{4}+\frac {1}{3} a^{2} x^{3}-a \,x^{2}-x \right )\) | \(54\) |
norman | \(a^{3} c^{3} x^{4}-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{5} a^{4} c^{3} x^{5}-\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{7} a^{6} c^{3} x^{7}-c^{3} a \,x^{2}\) | \(71\) |
risch | \(a^{3} c^{3} x^{4}-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{5} a^{4} c^{3} x^{5}-\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{7} a^{6} c^{3} x^{7}-c^{3} a \,x^{2}\) | \(71\) |
meijerg | \(\frac {c^{3} \left (-\frac {a x \left (120 a^{6} x^{6}+140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}+420 a x +840\right )}{840}-\ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{3} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{3} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{3} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c^{3} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{3} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{3} \ln \left (-a x +1\right )}{a}\) | \(319\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 70, normalized size = 1.35 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 70, normalized size = 1.35 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.03, size = 70, normalized size = 1.35 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 70, normalized size = 1.35 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 70, normalized size = 1.35 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________