Integrand size = 25, antiderivative size = 248 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^2}+\frac {5 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {5 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d^2} \]
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Time = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5932, 5901, 5903, 4267, 2317, 2438, 75, 106, 21, 94, 211, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {5 c^3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {5 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d^2}+\frac {13 b c^3 \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d^2}-\frac {b c^3}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c}{6 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 21
Rule 75
Rule 94
Rule 105
Rule 106
Rule 211
Rule 2317
Rule 2438
Rule 4267
Rule 5901
Rule 5903
Rule 5932
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {1}{3} \left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {1}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {3 c^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^2}-\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2} \\ & = \frac {5 b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (5 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 d^2}-\frac {\left (b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (5 b c^5\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (5 c^4\right ) \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 d^2}-\frac {\left (5 c^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{2 d^2}+\frac {\left (5 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d^2} \\ & = -\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d^2}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2}+\frac {\left (5 b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^2} \\ & = -\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {5 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^2}+\frac {5 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{2 d^2} \\ & = -\frac {b c^3}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arccosh}(c x))}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {13 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^2}+\frac {5 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {5 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {5 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d^2} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {\frac {4 a}{x^3}+\frac {24 a c^2}{x}-3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}+\frac {3 b c^3 \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}+\frac {3 b c^4 x \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}-\frac {2 b c^3}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 a c^4 x}{-1+c^2 x^2}+\frac {4 b \text {arccosh}(c x)}{x^3}+\frac {24 b c^2 \text {arccosh}(c x)}{x}+\frac {3 b c^3 \text {arccosh}(c x)}{-1+c x}+\frac {3 b c^3 \text {arccosh}(c x)}{1+c x}-\frac {26 b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+30 b c^3 \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-30 b c^3 \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+15 a c^3 \log (1-c x)-15 a c^3 \log (1+c x)-30 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+30 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{12 d^2} \]
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Time = 0.84 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\) | \(251\) |
default | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\) | \(251\) |
parts | \(\frac {a \left (-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b \,c^{3} \left (-\frac {15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-10 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {5 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {5 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\) | \(259\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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