Integrand size = 25, antiderivative size = 180 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {11}{5} b c d^3 \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{5} b c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{25} b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+b c d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.33, 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, 1624, 1813, 1634, 65, 211} \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+\frac {b c d^3 \sqrt {c^2 x^2-1} \arctan \left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 65
Rule 211
Rule 276
Rule 1624
Rule 1634
Rule 1813
Rule 5921
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^3 \left (-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6\right )}{5 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} \left (b c d^3\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {-1+c^2 x^2}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-5-15 c^2 x+5 c^4 x^2-c^6 x^3}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {11 c^2}{\sqrt {-1+c^2 x}}-\frac {5}{x \sqrt {-1+c^2 x}}+3 c^2 \sqrt {-1+c^2 x}-c^2 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {\left (b d^3 \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 {11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {b c d^3 \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 = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{25} d^3 \left (-\frac {25 a}{x}-75 a c^2 x+25 a c^4 x^3-5 a c^6 x^5+b c \sqrt {-1+c x} \sqrt {1+c x} \left (61-7 c^2 x^2+c^4 x^4\right )-\frac {5 b \left (5+15 c^2 x^2-5 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)}{x}-25 b c \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \]
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Time = 0.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99
method | result | size |
parts | \(-d^{3} a \left (\frac {c^{6} x^{5}}{5}-c^{4} x^{3}+3 c^{2} x +\frac {1}{x}\right )-d^{3} b c \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\) | \(178\) |
derivativedivides | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(181\) |
default | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\) | \(181\) |
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Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {5 \, a c^{6} d^{3} x^{6} - 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 50 \, b c d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 25 \, a d^{3} + 5 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x + 5 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{25 \, x} \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=- d^{3} \left (\int 3 a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int \left (- 3 a c^{4} x^{2}\right )\, dx + \int a c^{6} x^{4}\, dx + \int 3 b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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Time = 0.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \]
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