Optimal. Leaf size=163 \[ -\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {315 c^3 \text {ArcSin}(a x)}{8 a} \]
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Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6262, 683, 685,
655, 222} \begin {gather*} -\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 c^3 \sqrt {1-a^2 x^2} (1-a x)^3}{4 a}-\frac {21 c^3 \sqrt {1-a^2 x^2} (1-a x)^2}{4 a}-\frac {105 c^3 \sqrt {1-a^2 x^2} (1-a x)}{8 a}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {315 c^3 \text {ArcSin}(a x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 683
Rule 685
Rule 6262
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=\frac {\int \frac {(c-a c x)^6}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 \int \frac {(c-a c x)^4}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {63}{4} \int \frac {(c-a c x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{4} (105 c) \int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{8} \left (315 c^2\right ) \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{8} \left (315 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {315 c^3 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 45, normalized size = 0.28 \begin {gather*} -\frac {c^3 (1-a x)^{11/2} \, _2F_1\left (\frac {3}{2},\frac {11}{2};\frac {13}{2};\frac {1}{2} (1-a x)\right )}{11 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(520\) vs.
\(2(143)=286\).
time = 1.17, size = 521, normalized size = 3.20
method | result | size |
risch | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+67 a x -240\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{8 a \sqrt {-a^{2} x^{2}+1}}+\left (-\frac {315 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {32 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{3}\) | \(120\) |
default | \(-c^{3} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {6 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{a}-\frac {8 \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{a^{3}}+\frac {\frac {12 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+36 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{a^{2}}\right )\) | \(521\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.47, size = 233, normalized size = 1.43 \begin {gather*} -\frac {1}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + 3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3} x - \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {6 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{2} x + a} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a} - \frac {3 i \, c^{3} \arcsin \left (a x + 2\right )}{a} - \frac {339 \, c^{3} \arcsin \left (a x\right )}{8 \, a} - \frac {48 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x + a} + \frac {6 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3}}{a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 118, normalized size = 0.72 \begin {gather*} -\frac {496 \, a c^{3} x + 496 \, c^{3} - 630 \, {\left (a c^{3} x + c^{3}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a^{2} x + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c^{3} \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \left (- \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 103, normalized size = 0.63 \begin {gather*} -\frac {315 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {240 \, c^{3}}{a} - {\left (67 \, c^{3} + 2 \, {\left (a^{2} c^{3} x - 8 \, a c^{3}\right )} x\right )} x\right )} + \frac {64 \, c^{3}}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 166, normalized size = 1.02 \begin {gather*} \frac {32\,c^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {315\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {4\,a^3\,c^3}{{\left (-a^2\right )}^{3/2}}-\frac {67\,c^3\,x\,\sqrt {-a^2}}{8}-\frac {26\,a\,c^3}{\sqrt {-a^2}}+\frac {c^3\,x^3\,{\left (-a^2\right )}^{3/2}}{4}+\frac {2\,a^5\,c^3\,x^2}{{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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