Optimal. Leaf size=105 \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a} \]
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Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6138, 641, 195, 216} \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 6138
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1+a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+c^3 \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} \left (5 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{8} \left (5 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{16} \left (5 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 91, normalized size = 0.87 \[ \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (48 a^6 x^6+56 a^5 x^5-144 a^4 x^4-182 a^3 x^3+144 a^2 x^2+231 a x-48\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 115, normalized size = 1.10 \[ -\frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (48 \, a^{6} c^{3} x^{6} + 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} - 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} + 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 103, normalized size = 0.98 \[ \frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, {\left | a \right |}} - \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {48 \, c^{3}}{a} - {\left (231 \, c^{3} + 2 \, {\left (72 \, a c^{3} - {\left (91 \, a^{2} c^{3} + 4 \, {\left (18 \, a^{3} c^{3} - {\left (6 \, a^{5} c^{3} x + 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 183, normalized size = 1.74 \[ \frac {c^{3} a^{5} x^{6} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {3 c^{3} a^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{7}+\frac {3 c^{3} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{7 a}+\frac {c^{3} a^{4} x^{5} \sqrt {-a^{2} x^{2}+1}}{6}-\frac {13 c^{3} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{24}+\frac {11 c^{3} x \sqrt {-a^{2} x^{2}+1}}{16}+\frac {5 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 164, normalized size = 1.56 \[ \frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{5} - \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4} - \frac {13}{24} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{3} + \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} + \frac {11}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{7 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 100, normalized size = 0.95 \[ \frac {5\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {5\,c^3\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{24}+\frac {c^3\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{6}-\frac {c^3\,{\left (1-a^2\,x^2\right )}^{7/2}}{7\,a}-\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{16\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.98, size = 267, normalized size = 2.54 \[ \begin {cases} \frac {- \frac {c^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} + c^{3} \left (\begin {cases} \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 2 c^{3} \left (\begin {cases} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} + \frac {\operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 2 c^{3} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{6} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{16} + \frac {\operatorname {asin}{\left (a x \right )}}{16} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\c^{3} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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