Integrand size = 21, antiderivative size = 624 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}} \]
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Time = 0.71 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5959, 5883, 105, 12, 94, 211, 5909, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 (-d)^{5/2}}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b c^3 \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]
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Rule 12
Rule 94
Rule 105
Rule 211
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5883
Rule 5909
Rule 5959
Rule 5962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d x^4}-\frac {e (a+b \text {arccosh}(c x))}{d^2 x^2}+\frac {e^2 (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \text {arccosh}(c x)}{x^4} \, dx}{d}-\frac {e \int \frac {a+b \text {arccosh}(c x)}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{d+e x^2} \, dx}{d^2} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {(b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d}-\frac {(b c e) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2}+\frac {e^2 \int \left (\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {(b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {\left (b c^2 e\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 (-d)^{5/2}}-\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {1}{6} \left (-\frac {2 a}{d x^3}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{d x^2}-\frac {2 b \text {arccosh}(c x)}{d x^3}+\frac {6 e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {6 b c e \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 32.84 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.68
method | result | size |
parts | \(a \left (\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}\right )+\frac {b \left (8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x -48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}-8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{4} x^{3} d^{3}}\) | \(424\) |
derivativedivides | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {b \left (-8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x +48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )-24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(437\) |
default | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {b \left (-8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x +48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )-24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(437\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \]
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