Optimal. Leaf size=94 \[ -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5786, 454, 92, 205} \[ -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 454
Rule 5786
Rubi steps
\begin {align*} \int \frac {\left (d+e 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 {e \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c) \int \frac {-d-3 e 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 {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} \left (b c \left (c^2 d+6 e\right )\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 {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} \left (b c^2 \left (c^2 d+6 e\right )\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 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 {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 128, normalized size = 1.36 \[ \frac {\frac {-2 a \sqrt {c x-1} \sqrt {c x+1} \left (d+3 e x^2\right )+b c x^3 \sqrt {c^2 x^2-1} \left (c^2 d+6 e\right ) \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )+b c d x \left (c^2 x^2-1\right )}{\sqrt {c x-1} \sqrt {c x+1}}-2 b \cosh ^{-1}(c x) \left (d+3 e x^2\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 139, normalized size = 1.48 \[ \frac {2 \, {\left (b c^{3} d + 6 \, b c e\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b d + 3 \, b e\right )} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b c d x - 6 \, a e x^{2} - 2 \, a d - 2 \, {\left (3 \, b e x^{2} - {\left (b d + 3 \, b e\right )} x^{3} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, x^{3}} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 146, normalized size = 1.55 \[ -\frac {a e}{x}-\frac {d a}{3 x^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) e}{x}-\frac {d b \,\mathrm {arccosh}\left (c x \right )}{3 x^{3}}-\frac {c^{3} d b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b c d \sqrt {c x -1}\, \sqrt {c x +1}}{6 x^{2}}-\frac {c b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e}{\sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 85, normalized size = 0.90 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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