Integrand size = 25, antiderivative size = 171 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \]
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Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5936, 5916, 5569, 4267, 2317, 2438, 39, 40} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^3}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {2 b c x}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rule 39
Rule 40
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5916
Rule 5936
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^3} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d^3} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ & = \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.92 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {a}{\left (-1+c^2 x^2\right )^2}-\frac {2 a}{-1+c^2 x^2}-\frac {b \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{12 (1+c x)^2}+\frac {b \left ((2-c x) \sqrt {-1+c x} \sqrt {1+c x}+3 \text {arccosh}(c x)\right )}{12 (-1+c x)^2}+\frac {5}{4} b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-\frac {5}{4} b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )+2 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{4 d^3} \]
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Time = 0.79 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.89
method | result | size |
parts | \(-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\ln \left (x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(324\) |
derivativedivides | \(-\frac {a \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(326\) |
default | \(-\frac {a \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(326\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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