Optimal. Leaf size=64 \[ -\frac {\cosh ^{-1}(a+b x)}{x}-\frac {2 b \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\sqrt {1-a^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5996, 5963, 95,
211} \begin {gather*} -\frac {2 b \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\sqrt {1-a^2}}-\frac {\cosh ^{-1}(a+b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 211
Rule 5963
Rule 5996
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a+b x)}{x^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cosh ^{-1}(a+b x)}{x}+\text {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 )\\ &=-\frac {\cosh ^{-1}(a+b x)}{x}+2 \text {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 )\\ &=-\frac {\cosh ^{-1}(a+b x)}{x}-\frac {2 b \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\sqrt {1-a^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 83, normalized size = 1.30 \begin {gather*} -\frac {\cosh ^{-1}(a+b x)}{x}-\frac {i b \log \left (\frac {2 \left (\sqrt {-1+a+b x} \sqrt {1+a+b x}+\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}\right )}{b x}\right )}{\sqrt {1-a^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.60, size = 101, normalized size = 1.58
method | result | size |
derivativedivides | \(b \left (-\frac {\mathrm {arccosh}\left (b x +a \right )}{b x}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )}{\sqrt {\left (b x +a \right )^{2}-1}\, \left (-1+a \right ) \left (1+a \right )}\right )\) | \(101\) |
default | \(b \left (-\frac {\mathrm {arccosh}\left (b x +a \right )}{b x}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \sqrt {a^{2}-1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )}{\sqrt {\left (b x +a \right )^{2}-1}\, \left (-1+a \right ) \left (1+a \right )}\right )\) | \(101\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs.
\(2 (54) = 108\).
time = 0.39, size = 322, normalized size = 5.03 \begin {gather*} \left [\frac {\sqrt {a^{2} - 1} b x \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 )} x \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (a^{2} - {\left (a^{2} - 1\right )} x - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{{\left (a^{2} - 1\right )} x}, \frac {2 \, \sqrt {-a^{2} + 1} b x \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 )} x \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (a^{2} - {\left (a^{2} - 1\right )} x - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{{\left (a^{2} - 1\right )} x}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acosh}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 73, normalized size = 1.14 \begin {gather*} \frac {2 \, 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 )}{\sqrt {-a^{2} + 1}} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acosh}\left (a+b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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