Optimal. Leaf size=55 \[ \frac {2 (a x+1)^2}{a \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6123, 853, 669, 641, 216} \[ \frac {2 (a x+1)^2}{a \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 669
Rule 853
Rule 6123
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \, dx &=\int \frac {(1+a x)^2}{(1-a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (1+a x)^2}{a \sqrt {1-a^2 x^2}}-3 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 (1+a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a}-3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 (1+a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a}-\frac {3 \sin ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.71 \[ \frac {\sqrt {1-a^2 x^2} \left (1-\frac {4}{a x-1}\right )}{a}-\frac {3 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 65, normalized size = 1.18 \[ \frac {5 \, a x + 6 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 5\right )} - 5}{a^{2} x - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 63, normalized size = 1.15 \[ -\frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a} + \frac {8}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 1.44 \[ -\frac {a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {5}{a \sqrt {-a^{2} x^{2}+1}}+\frac {4 x}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 60, normalized size = 1.09 \[ -\frac {a x^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, \arcsin \left (a x\right )}{a} + \frac {5}{\sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 81, normalized size = 1.47 \[ \frac {\sqrt {1-a^2\,x^2}}{a}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {4\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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