Optimal. Leaf size=90 \[ -\frac {\left (2 a^2+1\right ) \cosh ^{-1}(a+b x)}{4 b^2}+\frac {3 a \sqrt {a+b x-1} \sqrt {a+b x+1}}{4 b^2}+\frac {1}{2} x^2 \cosh ^{-1}(a+b x)-\frac {x \sqrt {a+b x-1} \sqrt {a+b x+1}}{4 b} \]
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Rubi [A] time = 0.06, antiderivative size = 90, 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 = {5866, 5802, 90, 80, 52} \[ -\frac {\left (2 a^2+1\right ) \cosh ^{-1}(a+b x)}{4 b^2}+\frac {3 a \sqrt {a+b x-1} \sqrt {a+b x+1}}{4 b^2}+\frac {1}{2} x^2 \cosh ^{-1}(a+b x)-\frac {x \sqrt {a+b x-1} \sqrt {a+b x+1}}{4 b} \]
Antiderivative was successfully verified.
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Rule 52
Rule 80
Rule 90
Rule 5802
Rule 5866
Rubi steps
\begin {align*} \int x \cosh ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \cosh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \cosh ^{-1}(a+b x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )\\ &=-\frac {x \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b}+\frac {1}{2} x^2 \cosh ^{-1}(a+b x)-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\frac {1+2 a^2}{b^2}-\frac {3 a x}{b^2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )\\ &=\frac {3 a \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b^2}-\frac {x \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b}+\frac {1}{2} x^2 \cosh ^{-1}(a+b x)-\frac {\left (1+2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )}{4 b^2}\\ &=\frac {3 a \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b^2}-\frac {x \sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 b}-\frac {\left (1+2 a^2\right ) \cosh ^{-1}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \cosh ^{-1}(a+b x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 87, normalized size = 0.97 \[ \frac {-\left (2 a^2+1\right ) \log \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )+2 b^2 x^2 \cosh ^{-1}(a+b x)+(3 a-b x) \sqrt {a+b x-1} \sqrt {a+b x+1}}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 75, normalized size = 0.83 \[ \frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.91, size = 112, normalized size = 1.24 \[ \frac {1}{2} \, x^{2} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{4} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} + 1\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{2} {\left | b \right |}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 1.33 \[ \frac {x^{2} \mathrm {arccosh}\left (b x +a \right )}{2}-\frac {\mathrm {arccosh}\left (b x +a \right ) a^{2}}{2 b^{2}}-\frac {x \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{4 b}+\frac {3 a \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{4 b^{2}}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )}{4 b^{2} \sqrt {\left (b x +a \right )^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 151, normalized size = 1.68 \[ \frac {1}{2} \, x^{2} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {acosh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 104, normalized size = 1.16 \[ \begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{4 b^{2}} + \frac {x^{2} \operatorname {acosh}{\left (a + b x \right )}}{2} - \frac {x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{4 b} - \frac {\operatorname {acosh}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acosh}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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