Optimal. Leaf size=76 \[ b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )+b c d \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {14, 5921, 12,
471, 94, 211} \begin {gather*} c^2 (-d) x \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+b c d \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+b c d \sqrt {c x-1} \sqrt {c x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 94
Rule 211
Rule 471
Rule 5921
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d \left (-1-c^2 x^2\right )}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )-(b c d) \int \frac {-1-c^2 x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )+(b c d) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )+\left (b c^2 d\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}-c^2 d x \left (a+b \cosh ^{-1}(c x)\right )+b c d \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 110, normalized size = 1.45 \begin {gather*} -\frac {a d}{x}-a c^2 d x+b c d \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \cosh ^{-1}(c x)}{x}-b c^2 d x \cosh ^{-1}(c x)+\frac {b c d \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.95, size = 101, normalized size = 1.33
method | result | size |
derivativedivides | \(c \left (-a d \left (c x +\frac {1}{c x}\right )-b d \,\mathrm {arccosh}\left (c x \right ) c x -\frac {b d \,\mathrm {arccosh}\left (c x \right )}{c x}+b d \sqrt {c x -1}\, \sqrt {c x +1}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}\right )\) | \(101\) |
default | \(c \left (-a d \left (c x +\frac {1}{c x}\right )-b d \,\mathrm {arccosh}\left (c x \right ) c x -\frac {b d \,\mathrm {arccosh}\left (c x \right )}{c x}+b d \sqrt {c x -1}\, \sqrt {c x +1}-\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}\right )\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 66, normalized size = 0.87 \begin {gather*} -a c^{2} d x - {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d - \frac {a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 127, normalized size = 1.67 \begin {gather*} -\frac {a c^{2} d x^{2} - 2 \, b c d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} b c d x - {\left (b c^{2} + b\right )} d x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a d + {\left (b c^{2} d x^{2} - {\left (b c^{2} + b\right )} d x + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{x} \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*} - d \left (\int a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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