Integrand size = 25, antiderivative size = 250 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \]
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Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5932, 5936, 5916, 5569, 4267, 2317, 2438, 39, 40, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {6 c^2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^3}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {2 b c^3 x}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c^3 x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {b c}{2 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rule 12
Rule 39
Rule 40
Rule 105
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5916
Rule 5932
Rule 5936
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {(b c) \int \frac {4 c^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3}-\frac {\left (3 b c^3\right ) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c^3 x}{4 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\left (3 b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\left (2 b c^3\right ) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (3 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^3}-\frac {\left (4 b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (6 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (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 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.61 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {2 a}{x^2}+\frac {a c^2}{\left (-1+c^2 x^2\right )^2}-\frac {4 a c^2}{-1+c^2 x^2}-\frac {b c^2 \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{12 (-1+c x)^2}-\frac {b c^2 \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{12 (1+c x)^2}+\frac {2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+\frac {9}{4} b c^2 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-\frac {9}{4} b c^2 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+12 a c^2 \log (x)-6 a c^2 \log \left (1-c^2 x^2\right )+6 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{4 d^3} \]
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Time = 0.83 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.56
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\) | \(391\) |
default | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\) | \(391\) |
parts | \(-\frac {a \left (-\frac {c^{2}}{16 \left (c x +1\right )^{2}}-\frac {9 c^{2}}{16 \left (c x +1\right )}+\frac {3 c^{2} \ln \left (c x +1\right )}{2}+\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )-\frac {c^{2}}{16 \left (c x -1\right )^{2}}+\frac {9 c^{2}}{16 \left (c x -1\right )}+\frac {3 c^{2} \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \,c^{2} \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(406\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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