Integrand size = 25, antiderivative size = 239 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}+d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5919, 5882, 3799, 2221, 2317, 2438, 38, 54} \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}+d^3 \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))-\frac {1}{2} b d^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{36} b c d^3 x (c x-1)^{5/2} (c x+1)^{5/2}-\frac {7}{72} b c d^3 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac {19}{48} b c d^3 x \sqrt {c x-1} \sqrt {c x+1} \]
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Rule 38
Rule 54
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+d \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx+\frac {1}{6} \left (b c d^3\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx \\ & = \frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+d^2 \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx-\frac {1}{36} \left (5 b c d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx-\frac {1}{4} \left (b c d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx \\ & = -\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+d^3 \int \frac {a+b \text {arccosh}(c x)}{x} \, dx+\frac {1}{48} \left (5 b c d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx+\frac {1}{16} \left (3 b c d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx+\frac {1}{2} \left (b c d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx \\ & = \frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))-\frac {d^3 \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b}-\frac {1}{96} \left (5 b c d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{32} \left (3 b c d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b} \\ & = \frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}+d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-d^3 \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right ) \\ & = \frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}+d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {1}{2} \left (b d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ & = \frac {19}{48} b c d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{72} b c d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{36} b c d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}-\frac {19}{48} b d^3 \text {arccosh}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))+\frac {d^3 (a+b \text {arccosh}(c x))^2}{2 b}+d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d^3 \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {1}{144} d^3 \left (216 a c^2 x^2-108 a c^4 x^4+24 a c^6 x^6+33 b c x \sqrt {\frac {-1+c x}{1+c x}}+33 b c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+22 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}+22 b c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}}-4 b c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}}-4 b c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}}-108 b c x \sqrt {-1+c x} \sqrt {1+c x}-72 b \text {arccosh}(c x)^2-150 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+12 b \text {arccosh}(c x) \left (18 c^2 x^2-9 c^4 x^4+2 c^6 x^6-12 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-144 a \log (x)+72 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
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Time = 0.95 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.03
method | result | size |
parts | \(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (x \right )\right )-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {d^{3} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {25 b \,d^{3} \operatorname {arccosh}\left (c x \right )}{48}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}}{36}-\frac {11 d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{72}+\frac {25 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{48}\) | \(245\) |
derivativedivides | \(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (c x \right )\right )+\frac {25 b \,d^{3} \operatorname {arccosh}\left (c x \right )}{48}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}}{36}-\frac {11 d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{72}+\frac {25 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{48}+\frac {d^{3} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(247\) |
default | \(-d^{3} a \left (\frac {c^{6} x^{6}}{6}-\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}-\ln \left (c x \right )\right )+\frac {25 b \,d^{3} \operatorname {arccosh}\left (c x \right )}{48}+d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}}{36}-\frac {11 d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{72}+\frac {25 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{48}+\frac {d^{3} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(247\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=- d^{3} \left (\int \left (- \frac {a}{x}\right )\, dx + \int 3 a c^{2} x\, dx + \int \left (- 3 a c^{4} x^{3}\right )\, dx + \int a c^{6} x^{5}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int 3 b c^{2} x \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{3} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{5} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x} \,d x \]
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