Optimal. Leaf size=104 \[ -\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (11 a^2-5 a b x+4\right )}{18 b^3}+\frac {a \left (2 a^2+3\right ) \cosh ^{-1}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \cosh ^{-1}(a+b x)-\frac {x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{9 b} \]
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Rubi [A] time = 0.12, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5866, 5802, 100, 147, 52} \[ -\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (11 a^2-5 a b x+4\right )}{18 b^3}+\frac {a \left (2 a^2+3\right ) \cosh ^{-1}(a+b x)}{6 b^3}-\frac {x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{9 b}+\frac {1}{3} x^3 \cosh ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 52
Rule 100
Rule 147
Rule 5802
Rule 5866
Rubi steps
\begin {align*} \int x^2 \cosh ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \cosh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \cosh ^{-1}(a+b x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )\\ &=-\frac {x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{9 b}+\frac {1}{3} x^3 \cosh ^{-1}(a+b x)-\frac {1}{9} \operatorname {Subst}\left (\int \frac {\left (\frac {2+3 a^2}{b^2}-\frac {5 a x}{b^2}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )\\ &=-\frac {x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{9 b}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4+11 a^2-5 a b x\right )}{18 b^3}+\frac {1}{3} x^3 \cosh ^{-1}(a+b x)+\frac {\left (a \left (3+2 a^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,a+b x\right )}{6 b^3}\\ &=-\frac {x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{9 b}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4+11 a^2-5 a b x\right )}{18 b^3}+\frac {a \left (3+2 a^2\right ) \cosh ^{-1}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \cosh ^{-1}(a+b x)\\ \end {align*}
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Mathematica [A] time = 0.13, size = 101, normalized size = 0.97 \[ \frac {\left (6 a^3+9 a\right ) \log \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )-\sqrt {a+b x-1} \sqrt {a+b x+1} \left (11 a^2-5 a b x+2 b^2 x^2+4\right )+6 b^3 x^3 \cosh ^{-1}(a+b x)}{18 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 91, normalized size = 0.88 \[ \frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{18 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.85, size = 132, normalized size = 1.27 \[ \frac {1}{3} \, x^{3} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{18} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (x {\left (\frac {2 \, x}{b^{2}} - \frac {5 \, a}{b^{3}}\right )} + \frac {11 \, a^{2} b + 4 \, b}{b^{5}}\right )} + \frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{3} {\left | b \right |}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 207, normalized size = 1.99 \[ \frac {x^{3} \mathrm {arccosh}\left (b x +a \right )}{3}-\frac {x^{2} \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{9 b}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )}{3 b^{3} \sqrt {\left (b x +a \right )^{2}-1}}+\frac {5 \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, x a}{18 b^{2}}-\frac {11 \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{2}}{18 b^{3}}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 b^{3} \sqrt {\left (b x +a \right )^{2}-1}}-\frac {2 \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{9 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 212, normalized size = 2.04 \[ \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{4}} + \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \]
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 \[ \int x^2\,\mathrm {acosh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 170, normalized size = 1.63 \[ \begin {cases} \frac {a^{3} \operatorname {acosh}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{18 b^{3}} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{18 b^{2}} + \frac {a \operatorname {acosh}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {acosh}{\left (a + b x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{9 b} - \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} - 1}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acosh}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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