Optimal. Leaf size=71 \[ -\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5662, 103, 12, 92, 205} \[ -\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 205
Rule 5662
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{3} (b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} (b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} \left (b c^4\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 x^3}+\frac {1}{6} b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 101, normalized size = 1.42 \[ -\frac {a}{3 x^3}+\frac {b c^3 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \cosh ^{-1}(c x)}{3 x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 100, normalized size = 1.41 \[ \frac {2 \, b c^{3} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b c x + 2 \, {\left (b x^{3} - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, a}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 1.15 \[ -\frac {a}{3 x^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 x^{3}}-\frac {c^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b c \sqrt {c x -1}\, \sqrt {c x +1}}{6 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 52, normalized size = 0.73 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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