Optimal. Leaf size=90 \[ \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {5}{6} b c^3 d \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 90, 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,
465, 94, 211} \begin {gather*} \frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {5}{6} b c^3 d \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 94
Rule 211
Rule 465
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^4} \, dx &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-(b c) \int \frac {d \left (-1+3 c^2 x^2\right )}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c d) \int \frac {-1+3 c^2 x^2}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{6} \left (5 b c^3 d\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{6} \left (5 b c^4 d\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {5}{6} b c^3 d \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 127, normalized size = 1.41 \begin {gather*} -\frac {a d}{3 x^3}+\frac {a c^2 d}{x}+\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {b d \cosh ^{-1}(c x)}{3 x^3}+\frac {b c^2 d \cosh ^{-1}(c x)}{x}-\frac {5 b c^3 d \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.99, size = 119, normalized size = 1.32
method | result | size |
derivativedivides | \(c^{3} \left (-a d \left (-\frac {1}{c x}+\frac {1}{3 c^{3} x^{3}}\right )+\frac {b d \,\mathrm {arccosh}\left (c x \right )}{c x}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {5 b d \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{6 c^{2} x^{2}}\right )\) | \(119\) |
default | \(c^{3} \left (-a d \left (-\frac {1}{c x}+\frac {1}{3 c^{3} x^{3}}\right )+\frac {b d \,\mathrm {arccosh}\left (c x \right )}{c x}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {5 b d \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{6 c^{2} x^{2}}\right )\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 89, normalized size = 0.99 \begin {gather*} {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d - \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 d + \frac {a c^{2} d}{x} - \frac {a d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 146, normalized size = 1.62 \begin {gather*} -\frac {10 \, b c^{3} d x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, a c^{2} d x^{2} + 2 \, {\left (3 \, b c^{2} - b\right )} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} b c d x + 2 \, a d - 2 \, {\left (3 \, b c^{2} d x^{2} - {\left (3 \, b c^{2} - b\right )} d x^{3} - b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, x^{3}} \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 \left (- \frac {a}{x^{4}}\right )\, dx + \int \frac {a c^{2}}{x^{2}}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {b c^{2} \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, 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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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