Optimal. Leaf size=131 \[ \frac {51 \sin ^{-1}(a x)}{8 a^4}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {9 (2-3 a x) \sqrt {1-a^2 x^2}}{8 a^4}+\frac {27 \sqrt {1-a^2 x^2}}{4 a^4}+\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.69, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6124, 1633, 1593, 12, 852, 1635, 1815, 27, 743, 641, 216} \[ -\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}+\frac {9 (2-3 a x) \sqrt {1-a^2 x^2}}{8 a^4}+\frac {27 \sqrt {1-a^2 x^2}}{4 a^4}+\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {51 \sin ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 216
Rule 641
Rule 743
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1815
Rule 6124
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1-a x)^2}{(1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=a \int \frac {\sqrt {1-a^2 x^2} \left (\frac {x^3}{a}-x^4\right )}{(1+a x)^2} \, dx\\ &=a \int \frac {\left (\frac {1}{a}-x\right ) x^3 \sqrt {1-a^2 x^2}}{(1+a x)^2} \, dx\\ &=a^2 \int \frac {x^3 \left (1-a^2 x^2\right )^{3/2}}{a^2 (1+a x)^3} \, dx\\ &=\int \frac {x^3 \left (1-a^2 x^2\right )^{3/2}}{(1+a x)^3} \, dx\\ &=\int \frac {x^3 (1-a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}-\int \frac {(1-a x)^2 \left (-\frac {3}{a^3}+\frac {x}{a^2}-\frac {x^2}{a}\right )}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {\int \frac {\frac {12}{a}-28 x+27 a x^2-12 a^2 x^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {\int \frac {-36 a+108 a^2 x-81 a^3 x^2}{\sqrt {1-a^2 x^2}} \, dx}{12 a^4}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {\int -\frac {9 a (-2+3 a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{12 a^4}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {3 \int \frac {(-2+3 a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a^3}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {9 (2-3 a x) \sqrt {1-a^2 x^2}}{8 a^4}-\frac {3 \int \frac {-17 a^2+18 a^3 x}{\sqrt {1-a^2 x^2}} \, dx}{8 a^5}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {27 \sqrt {1-a^2 x^2}}{4 a^4}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {9 (2-3 a x) \sqrt {1-a^2 x^2}}{8 a^4}+\frac {51 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}\\ &=\frac {(1-a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {27 \sqrt {1-a^2 x^2}}{4 a^4}+\frac {x^2 \sqrt {1-a^2 x^2}}{a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}+\frac {9 (2-3 a x) \sqrt {1-a^2 x^2}}{8 a^4}+\frac {51 \sin ^{-1}(a x)}{8 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 70, normalized size = 0.53 \[ \frac {51 \sin ^{-1}(a x)}{8 a^4}+\sqrt {1-a^2 x^2} \left (\frac {4}{a^4 (a x+1)}+\frac {6}{a^4}-\frac {19 x}{8 a^3}+\frac {x^2}{a^2}-\frac {x^3}{4 a}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 92, normalized size = 0.70 \[ \frac {80 \, a x - 102 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right )} \sqrt {-a^{2} x^{2} + 1} + 80}{8 \, {\left (a^{5} x + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 235, normalized size = 1.79 \[ \frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{3}}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{7} \left (x +\frac {1}{a}\right )^{3}}+\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{6} \left (x +\frac {1}{a}\right )^{2}}+\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a^{4}}+\frac {6 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{a^{3}}+\frac {6 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{a^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.43, size = 215, normalized size = 1.64 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{6} x^{2} + 2 \, a^{5} x + a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (a^{5} x + a^{4}\right )}} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{3}} - \frac {3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} x}{2 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{3}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}} + \frac {3 i \, \arcsin \left (a x + 2\right )}{2 \, a^{4}} + \frac {63 \, \arcsin \left (a x\right )}{8 \, a^{4}} - \frac {3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3}}{a^{4}} + \frac {9 \, \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 154, normalized size = 1.18 \[ \frac {51\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^3\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{{\left (-a^2\right )}^{3/2}}-\frac {4}{a^2\,\sqrt {-a^2}}-\frac {19\,x\,\sqrt {-a^2}}{8\,a^3}+\frac {a^2\,x^2}{{\left (-a^2\right )}^{3/2}}+\frac {x^3\,{\left (-a^2\right )}^{3/2}}{4\,a^3}\right )}{\sqrt {-a^2}}-\frac {4\,\sqrt {1-a^2\,x^2}}{a^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________