Optimal. Leaf size=73 \[ -8 c^3 x+\frac {2 c^3 (1-a x)^2}{a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}+\frac {16 c^3 \log (1+a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {6264, 45}
\begin {gather*} \frac {c^3 (1-a x)^4}{4 a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {2 c^3 (1-a x)^2}{a}+\frac {16 c^3 \log (a x+1)}{a}-8 c^3 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6264
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int \frac {(1-a x)^4}{1+a x} \, dx\\ &=c^3 \int \left (-8-4 (1-a x)-2 (1-a x)^2-(1-a x)^3+\frac {16}{1+a x}\right ) \, dx\\ &=-8 c^3 x+\frac {2 c^3 (1-a x)^2}{a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}+\frac {16 c^3 \log (1+a x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 48, normalized size = 0.66 \begin {gather*} \frac {c^3 \left (35-180 a x+66 a^2 x^2-20 a^3 x^3+3 a^4 x^4+192 \log (1+a x)\right )}{12 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.93, size = 42, normalized size = 0.58
method | result | size |
default | \(c^{3} \left (\frac {a^{3} x^{4}}{4}-\frac {5 a^{2} x^{3}}{3}+\frac {11 a \,x^{2}}{2}-15 x +\frac {16 \ln \left (a x +1\right )}{a}\right )\) | \(42\) |
risch | \(\frac {a^{3} c^{3} x^{4}}{4}-\frac {5 a^{2} c^{3} x^{3}}{3}+\frac {11 a \,c^{3} x^{2}}{2}-15 x \,c^{3}+\frac {16 c^{3} \ln \left (a x +1\right )}{a}\) | \(53\) |
norman | \(\frac {-15 x \,c^{3}-\frac {19}{2} a \,c^{3} x^{2}+\frac {23}{6} a^{2} c^{3} x^{3}-\frac {17}{12} a^{3} c^{3} x^{4}+\frac {1}{4} c^{3} a^{4} x^{5}}{a x +1}+\frac {16 c^{3} \ln \left (a x +1\right )}{a}\) | \(73\) |
meijerg | \(\frac {c^{3} \left (-\frac {a x \left (-3 a^{4} x^{4}+5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{12 \left (a x +1\right )}+5 \ln \left (a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-\frac {a x \left (-2 a^{2} x^{2}+6 a x +12\right )}{4 \left (a x +1\right )}+3 \ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {a x \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}+\frac {2 c^{3} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (-\frac {a x}{a x +1}+\ln \left (a x +1\right )\right )}{a}+\frac {c^{3} x}{a x +1}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 52, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {5}{3} \, a^{2} c^{3} x^{3} + \frac {11}{2} \, a c^{3} x^{2} - 15 \, c^{3} x + \frac {16 \, c^{3} \log \left (a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 57, normalized size = 0.78 \begin {gather*} \frac {3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log \left (a x + 1\right )}{12 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 56, normalized size = 0.77 \begin {gather*} \frac {a^{3} c^{3} x^{4}}{4} - \frac {5 a^{2} c^{3} x^{3}}{3} + \frac {11 a c^{3} x^{2}}{2} - 15 c^{3} x + \frac {16 c^{3} \log {\left (a x + 1 \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 82, normalized size = 1.12 \begin {gather*} \frac {{\left (3 \, c^{3} - \frac {32 \, c^{3}}{a x + 1} + \frac {144 \, c^{3}}{{\left (a x + 1\right )}^{2}} - \frac {384 \, c^{3}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}^{4}}{12 \, a} - \frac {16 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.79, size = 52, normalized size = 0.71 \begin {gather*} \frac {11\,a\,c^3\,x^2}{2}-15\,c^3\,x-\frac {5\,a^2\,c^3\,x^3}{3}+\frac {a^3\,c^3\,x^4}{4}+\frac {16\,c^3\,\ln \left (a\,x+1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________