Integrand size = 19, antiderivative size = 94 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5956, 465, 94, 211} \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {d (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{x}+\frac {1}{6} b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right ) \left (c^2 d+6 e\right )+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]
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Rule 94
Rule 211
Rule 465
Rule 5956
Rubi steps \begin{align*} \text {integral}& = -\frac {d (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{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 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{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 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{x}+\frac {1}{6} \left (b c^2 \left (c^2 d+6 e\right )\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 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {e (a+b \text {arccosh}(c x))}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {-2 b \left (d+3 e x^2\right ) \text {arccosh}(c x)+\frac {b c d x \left (-1+c^2 x^2\right )-2 a \sqrt {-1+c x} \sqrt {1+c x} \left (d+3 e x^2\right )+b c \left (c^2 d+6 e\right ) x^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}}{6 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.50
method | result | size |
parts | \(a \left (-\frac {e}{x}-\frac {d}{3 x^{3}}\right )+b \,c^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right ) e}{c^{3} x}-\frac {\operatorname {arccosh}\left (c x \right ) d}{3 c^{3} x^{3}}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d \,x^{2}+6 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e \,c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{2} d \right )}{6 c^{4} \sqrt {c^{2} x^{2}-1}\, x^{2}}\right )\) | \(141\) |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arccosh}\left (c x \right ) e}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d \,x^{2}+6 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e \,c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{2} d \right )}{6 \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}}\right )}{c^{2}}\right )\) | \(154\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arccosh}\left (c x \right ) e}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{4} d \,x^{2}+6 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e \,c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\, c^{2} d \right )}{6 \sqrt {c^{2} x^{2}-1}\, c^{2} x^{2}}\right )}{c^{2}}\right )\) | \(154\) |
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Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.48 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {2 \, {\left (b c^{3} d + 6 \, b c e\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b d + 3 \, b e\right )} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b c d x - 6 \, a e x^{2} - 2 \, a d - 2 \, {\left (3 \, b e x^{2} - {\left (b d + 3 \, b e\right )} x^{3} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, x^{3}} \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^4} \,d x \]
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