Integrand size = 25, antiderivative size = 200 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+b c^2 d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))+b c^2 d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)+\frac {1}{4} b c^3 d^2 x \sqrt {c x-1} \sqrt {c x+1}-\frac {b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \]
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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^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\left (2 c^2 d\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^2\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x^2} \, dx \\ & = -\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} \left (b c d^2\right ) \int 3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \, dx-\left (2 c^2 d^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx-\left (b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx \\ & = -\frac {1}{2} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (2 c^2 d^2\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}{2} \left (b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{2} \left (3 b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx \\ & = \frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}+\frac {1}{2} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}+\frac {\left (4 c^2 d^2\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}{4} \left (3 b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\left (2 c^2 d^2\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 {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\left (b c^2 d^2\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 {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+b c^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {d^2 \left (-2 a+2 a c^4 x^4+2 b c x \sqrt {-1+c x} \sqrt {1+c x}-b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-4 b c^2 x^2 \text {arccosh}(c x)^2-2 b c^2 x^2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+2 b \text {arccosh}(c x) \left (-1+c^4 x^4-4 c^2 x^2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-8 a c^2 x^2 \log (x)+4 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{4 x^2} \]
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Time = 0.90 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}-2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \operatorname {arccosh}\left (c x \right )^{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(206\) |
default | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}-2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \operatorname {arccosh}\left (c x \right )^{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(206\) |
parts | \(d^{2} a \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )\right )+d^{2} b \,c^{2} \operatorname {arccosh}\left (c x \right )^{2}+\frac {d^{2} b \,c^{4} \operatorname {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \,c^{3} d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b \,c^{2} d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b \,c^{2}}{2}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{2 x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 x^{2}}-2 d^{2} b \,c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \,c^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\) | \(212\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \left (- \frac {2 a c^{2}}{x}\right )\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{4} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \]
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