Optimal. Leaf size=106 \[ -\frac {a b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{3/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x}-\frac {\cosh ^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 106, 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, 96, 93, 205} \[ -\frac {a b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{3/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x}-\frac {\cosh ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 205
Rule 5802
Rule 5866
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a+b x)}{x^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cosh ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\cosh ^{-1}(a+b x)}{2 x^2}+\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x\right )}{2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\cosh ^{-1}(a+b x)}{2 x^2}+\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{b}-\frac {a}{b}-\left (\frac {1}{b}-\frac {a}{b}\right ) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {-1+a+b x}}\right )}{1-a^2}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\cosh ^{-1}(a+b x)}{2 x^2}-\frac {a b^2 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 136, normalized size = 1.28 \[ \frac {-\cosh ^{-1}(a+b x)+\frac {b x \left (-\sqrt {a+b x-1} \sqrt {a+b x+1}+\frac {i a b x \log \left (\frac {4 i \sqrt {1-a^2} \left (-i \sqrt {1-a^2} \sqrt {a+b x-1} \sqrt {a+b x+1}+a^2+a b x-1\right )}{a b^2 x}\right )}{\sqrt {1-a^2}}\right )}{a^2-1}}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 460, normalized size = 4.34 \[ \left [\frac {\sqrt {a^{2} - 1} a b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - {\left (a^{2} - 1\right )} b^{2} x^{2} + {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} b x - {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, -\frac {2 \, \sqrt {-a^{2} + 1} a b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) + {\left (a^{2} - 1\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} b x + {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 170, normalized size = 1.60 \[ -{\left (\frac {a b \arctan \left (-\frac {x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{2} - 1\right )} \sqrt {-a^{2} + 1}} - \frac {{\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a b + a^{2} {\left | b \right |} - {\left | b \right |}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{2} - a^{2} + 1\right )} {\left (a^{2} - 1\right )}}\right )} b - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 181, normalized size = 1.71 \[ -\frac {\mathrm {arccosh}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a}{2 \sqrt {a^{2}-1}\, \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}-\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{2}}{2 x \left (a^{2}-1\right ) \left (1+a \right ) \left (a -1\right )}+\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{2 x \left (a^{2}-1\right ) \left (1+a \right ) \left (a -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\mathrm {acosh}\left (a+b\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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