Integrand size = 21, antiderivative size = 489 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d} \]
[Out]
Time = 0.66 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5959, 5882, 3799, 2221, 2317, 2438, 5962, 5681} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d}+\frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {\log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d} \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5681
Rule 5882
Rule 5959
Rule 5962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d x}-\frac {e x (a+b \text {arccosh}(c x))}{d \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{d}-\frac {e \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{d} \\ & = -\frac {\text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b d}-\frac {e \int \left (-\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b d}-\frac {2 \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 d}+\frac {\sqrt {e} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d}-\frac {\sqrt {e} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{2 b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}-\frac {\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 )}{d}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 d}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}+\frac {b \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 )}{2 d}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{b d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=-\frac {-2 b \text {arccosh}(c x)^2-2 b \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 a \log (x)+a \log \left (d+e x^2\right )+b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{2 d} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.60 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.78
method | result | size |
parts | \(-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d}+\frac {a \ln \left (x \right )}{d}+b \left (-\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}}{4 d}\right )\) | \(381\) |
derivativedivides | \(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}\) | \(393\) |
default | \(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}\) | \(393\) |
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \]
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,\left (e\,x^2+d\right )} \,d x \]
[In]
[Out]