Optimal. Leaf size=85 \[ -\frac {(1+a x)^2}{2 a^2}+\frac {(1+a x) \left (1+2 \sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}+\frac {2 \log (1+a x)}{a^2}+\frac {4 \log \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2} \]
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Rubi [A]
time = 0.28, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6472, 1661,
1607, 815, 266} \begin {gather*} -\frac {(a x+1)^2}{2 a^2}+\frac {\left (2 \sqrt {\frac {1-a x}{a x+1}}+1\right ) (a x+1)}{a^2}+\frac {2 \log (a x+1)}{a^2}+\frac {4 \log \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 815
Rule 1607
Rule 1661
Rule 6472
Rubi steps
\begin {align*} \int e^{2 \text {sech}^{-1}(a x)} x \, dx &=\int x \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2 \, dx\\ &=\frac {4 \text {Subst}\left (\int \frac {x (1+x)^3}{(-1+x) \left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}-\frac {\text {Subst}\left (\int \frac {-12 x-4 x^2}{(-1+x) \left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}-\frac {\text {Subst}\left (\int \frac {(-12-4 x) x}{(-1+x) \left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}+\frac {(1+a x) \left (1+2 \sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}+\frac {\text {Subst}\left (\int \frac {8+8 x}{(-1+x) \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}+\frac {(1+a x) \left (1+2 \sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}+\frac {\text {Subst}\left (\int \left (\frac {8}{-1+x}-\frac {8 x}{1+x^2}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}+\frac {(1+a x) \left (1+2 \sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}+\frac {4 \log \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}-\frac {4 \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac {(1+a x)^2}{2 a^2}+\frac {(1+a x) \left (1+2 \sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}+\frac {2 \log (1+a x)}{a^2}+\frac {4 \log \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 89, normalized size = 1.05 \begin {gather*} \frac {-a^2 x^2+4 \sqrt {\frac {1-a x}{1+a x}} (1+a x)+8 \log (x)-4 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 98, normalized size = 1.15
method | result | size |
default | \(\frac {-\frac {a^{2} x^{2}}{2}+\ln \left (x \right )}{a^{2}}-\frac {2 \sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (-\sqrt {-a^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a \sqrt {-a^{2} x^{2}+1}}+\frac {\ln \left (x \right )}{a^{2}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 124, normalized size = 1.46 \begin {gather*} -\frac {a^{2} x^{2} - 4 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 2 \, \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 2 \, \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 4 \, \log \left (x\right )}{2 \, a^{2}} \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*} \frac {\int \frac {2}{x}\, dx + \int \left (- a^{2} x\right )\, dx + \int 2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.69, size = 56, normalized size = 0.66 \begin {gather*} \frac {2\,x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{a}-\frac {2\,\mathrm {acosh}\left (\frac {1}{a\,x}\right )}{a^2}-\frac {x^2}{2}-\frac {2\,\ln \left (\frac {1}{x}\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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