Optimal. Leaf size=31 \[ \frac {e^{2 \cosh ^{-1}(a+b x)}}{4 b}-\frac {\cosh ^{-1}(a+b x)}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5897, 2282, 12, 14} \[ \frac {e^{2 \cosh ^{-1}(a+b x)}}{4 b}-\frac {\cosh ^{-1}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2282
Rule 5897
Rubi steps
\begin {align*} \int e^{\cosh ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int e^x \sinh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{2 x} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{x} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x}+x\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{2 b}\\ &=\frac {e^{2 \cosh ^{-1}(a+b x)}}{4 b}-\frac {\cosh ^{-1}(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 69, normalized size = 2.23 \[ \frac {(a+b x) \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )-\log \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )}{2 b} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.53, size = 66, normalized size = 2.13 \[ \frac {b^{2} x^{2} + 2 \, a b x + \sqrt {b x + a + 1} {\left (b x + a\right )} \sqrt {b x + a - 1} + \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 151, normalized size = 4.87 \[ \frac {1}{2} \, b x^{2} + a x + \frac {\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} + 2 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right ) + 2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 4 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 147, normalized size = 4.74 \[ \frac {b \,x^{2}}{2}+a x +\frac {\sqrt {b x +a -1}\, \left (b x +a +1\right )^{\frac {3}{2}}}{2 b}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{2 b}-\frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {\frac {\left (a -1\right ) b}{2}+\frac {b \left (1+a \right )}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (\left (a -1\right ) b +b \left (1+a \right )\right ) x +\left (a -1\right ) \left (1+a \right )}\right )}{2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 143, normalized size = 4.61 \[ \frac {1}{2} \, b x^{2} + a x - \frac {a^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x + \frac {{\left (a^{2} - 1\right )} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 79, normalized size = 2.55 \[ a\,x+\frac {b\,x^2}{2}-\frac {\ln \left (a+\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}+b\,x\right )}{2\,b}+\frac {x\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2}+\frac {a\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2\,b} \]
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 \[ \int \left (a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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