Integrand size = 25, antiderivative size = 135 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {5}{3} b c d^2 \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{9} b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))+b c d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {276, 5921, 12, 534, 1265, 911, 1167, 211} \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-2 c^2 d^2 x (a+b \text {arccosh}(c x))-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+\frac {b c d^2 \sqrt {c^2 x^2-1} \arctan \left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 211
Rule 276
Rule 534
Rule 911
Rule 1167
Rule 1265
Rule 5921
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^2 \left (-3-6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} \left (b c d^2\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-3-6 c^2 x+c^4 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-8-4 x^2+x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac {3}{\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {b c d^2 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d^2 \left (-9 a-18 a c^2 x^2+3 a c^4 x^4+16 b c x \sqrt {-1+c x} \sqrt {1+c x}-b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+3 b \left (-3-6 c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)-9 b c x \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{9 x} \]
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Time = 0.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.03
method | result | size |
parts | \(d^{2} a \left (\frac {c^{4} x^{3}}{3}-2 c^{2} x -\frac {1}{x}\right )+d^{2} b c \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\) | \(139\) |
derivativedivides | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(141\) |
default | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(141\) |
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Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.49 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {3 \, a c^{4} d^{2} x^{4} - 18 \, a c^{2} d^{2} x^{2} + 18 \, b c d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 9 \, a d^{2} + 3 \, {\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, x} \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=d^{2} \left (\int \left (- 2 a c^{2}\right )\, dx + \int \frac {a}{x^{2}}\, dx + \int a c^{4} x^{2}\, dx + \int \left (- 2 b c^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{2} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^2} \,d x \]
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