Integrand size = 21, antiderivative size = 634 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3} \]
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Time = 0.85 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5959, 5883, 97, 5882, 3799, 2221, 2317, 2438, 5957, 533, 385, 214, 5962, 5681} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{d^3}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}-\frac {2 e \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d^3}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b c e \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 d^{5/2} \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^2 x} \]
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Rule 97
Rule 214
Rule 385
Rule 533
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5681
Rule 5882
Rule 5883
Rule 5957
Rule 5959
Rule 5962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d^2 x^3}-\frac {2 e (a+b \text {arccosh}(c x))}{d^3 x}+\frac {e^2 x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^2}+\frac {2 e^2 x (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \text {arccosh}(c x)}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx}{d^2} \\ & = -\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d^2}+\frac {(2 e) \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^3}+\frac {(b c e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 d^2}+\frac {\left (2 e^2\right ) \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^3} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {(4 e) \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^3}-\frac {e^{3/2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{d^3}+\frac {e^{3/2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{d^3}+\frac {\left (b c e \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {e (a+b \text {arccosh}(c x))^2}{b d^3}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {(2 e) \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^3}-\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{d^3}+\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{d^3}+\frac {\left (b c e \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {e^{3/2} \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 )}{d^3}-\frac {e^{3/2} \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 )}{d^3}+\frac {e^{3/2} \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 )}{d^3}+\frac {e^{3/2} \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 )}{d^3} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^2 x}-\frac {a+b \text {arccosh}(c x)}{2 d^2 x^2}-\frac {e (a+b \text {arccosh}(c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 e (a+b \text {arccosh}(c x))^2}{b d^3}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {e (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^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{d^3}+\frac {b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{d^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.55 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {-\frac {2 a d}{x^2}-\frac {2 a d e}{d+e x^2}-8 a e \log (x)+4 a e \log \left (d+e x^2\right )+b \left (\frac {2 d \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}-4 e \text {arccosh}(c x)^2-4 e \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+4 e \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+4 e \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+i \sqrt {d} e \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+i \sqrt {d} e \left (-\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+4 e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.61 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {a e \ln \left (e \,x^{2}+d \right )}{d^{3}}-\frac {a e}{2 d^{2} \left (e \,x^{2}+d \right )}-\frac {a}{2 d^{2} x^{2}}-\frac {2 a e \ln \left (x \right )}{d^{3}}+b \,c^{2} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 c^{2} x^{2} \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 d^{3} c^{2} \left (c^{2} d +e \right )}+\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} 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 )}{2 d^{3} c^{2}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} c^{2}}+\frac {e^{2} \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 )}{2 d^{3} c^{2}}\right )\) | \(635\) |
derivativedivides | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}+\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} 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 )}{2 c^{6} d^{3}}+\frac {e^{2} \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 )}{2 c^{6} d^{3}}\right )\right )\) | \(664\) |
default | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d x -\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} e \,x^{3}+c^{4} d \,x^{2}+c^{4} e \,x^{4}+\operatorname {arccosh}\left (c x \right ) c^{2} d +2 \,\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {\sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, e \,\operatorname {arctanh}\left (\frac {4 c^{2} d +2 e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}-\frac {2 e \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{6} d^{3}}+\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} 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 )}{2 c^{6} d^{3}}+\frac {e^{2} \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 )}{2 c^{6} d^{3}}\right )\right )\) | \(664\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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