Integrand size = 25, antiderivative size = 267 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}-3 c^2 d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {3}{2} b c^2 d^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5920, 99, 12, 38, 54, 5919, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}-3 c^2 d^3 \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))+\frac {3}{2} b c^2 d^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+\frac {3}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c x-1} \sqrt {c x+1}+\frac {b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \]
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Rule 12
Rule 38
Rule 54
Rule 99
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rule 5920
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2} \, dx \\ & = \frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx-\frac {1}{2} \left (b c d^3\right ) \int 5 c^2 (-1+c x)^{3/2} (1+c x)^{3/2} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx \\ & = \frac {3}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (5 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx \\ & = -\frac {33}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (3 c^2 d^3\right ) \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}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {33}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}+\frac {\left (6 c^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 {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}-3 c^2 d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\left (3 c^2 d^3\right ) \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 {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}-3 c^2 d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} \left (3 b c^2 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 {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \text {arccosh}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))^2}{2 b}-3 c^2 d^3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {3}{2} b c^2 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 = 0.46 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {d^3 \left (-16 a+48 a c^4 x^4-8 a c^6 x^6+3 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}+3 b c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}}+2 b c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}}+2 b c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}}+16 b c x \sqrt {-1+c x} \sqrt {1+c x}-24 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-48 b c^2 x^2 \text {arccosh}(c x)^2-42 b c^2 x^2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )-8 b \text {arccosh}(c x) \left (2-6 c^4 x^4+c^6 x^6+12 c^2 x^2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-96 a c^2 x^2 \log (x)+48 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{32 x^2} \]
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Time = 1.00 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(c^{2} \left (-d^{3} a \left (\frac {c^{4} x^{4}}{4}-\frac {3 c^{2} x^{2}}{2}+3 \ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-\frac {d^{3} b}{2}-\frac {21 b \,d^{3} \operatorname {arccosh}\left (c x \right )}{32}-3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {3 d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}-\frac {21 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {3 d^{3} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\) | \(258\) |
default | \(c^{2} \left (-d^{3} a \left (\frac {c^{4} x^{4}}{4}-\frac {3 c^{2} x^{2}}{2}+3 \ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-\frac {d^{3} b}{2}-\frac {21 b \,d^{3} \operatorname {arccosh}\left (c x \right )}{32}-3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {3 d^{3} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}-\frac {21 b c \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {3 d^{3} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\) | \(258\) |
parts | \(-d^{3} a \left (\frac {c^{6} x^{4}}{4}-\frac {3 c^{4} x^{2}}{2}+\frac {1}{2 x^{2}}+3 c^{2} \ln \left (x \right )\right )-\frac {d^{3} b \,c^{2}}{2}-3 d^{3} b \,c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {3 d^{3} b \,c^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \,c^{5} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3}}{16}-\frac {21 b \,c^{3} d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}+\frac {d^{3} b c \sqrt {c x +1}\, \sqrt {c x -1}}{2 x}-\frac {21 b \,c^{2} d^{3} \operatorname {arccosh}\left (c x \right )}{32}-\frac {3 d^{3} b \,c^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{3} b \,\operatorname {arccosh}\left (c x \right )}{2 x^{2}}-\frac {d^{3} b \,c^{6} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}+\frac {3 d^{3} b \,c^{4} \operatorname {arccosh}\left (c x \right ) x^{2}}{2}\) | \(264\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=- d^{3} \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int \left (- 3 a c^{4} x\right )\, dx + \int a c^{6} x^{3}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b c^{4} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \]
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