Optimal. Leaf size=41 \[ \frac {(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5864, 5654, 74} \[ \frac {(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 74
Rule 5654
Rule 5864
Rubi steps
\begin {align*} \int \cosh ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{b}+\frac {(a+b x) \cosh ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 1.37 \[ x \cosh ^{-1}(a+b x)-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}-2 a \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 57, normalized size = 1.39 \[ \frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.94, size = 93, normalized size = 2.27 \[ -b {\left (\frac {a \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.88 \[ \frac {\left (b x +a \right ) \mathrm {arccosh}\left (b x +a \right )-\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 30, normalized size = 0.73 \[ \frac {{\left (b x + a\right )} \operatorname {arcosh}\left (b x + a\right ) - \sqrt {{\left (b x + a\right )}^{2} - 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.00, size = 266, normalized size = 6.49 \[ x\,\mathrm {acosh}\left (a+b\,x\right )-\frac {\frac {4\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{b\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {4\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}-\frac {8\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\sqrt {a-1}\,\sqrt {a+1}}{b\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}}{\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+1}+\frac {4\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 46, normalized size = 1.12 \[ \begin {cases} \frac {a \operatorname {acosh}{\left (a + b x \right )}}{b} + x \operatorname {acosh}{\left (a + b x \right )} - \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acosh}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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