Integrand size = 21, antiderivative size = 772 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\frac {b c \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 {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\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^3}-\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^3}-\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^3}-\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^3} \]
[Out]
Time = 0.95 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5959, 5882, 3799, 2221, 2317, 2438, 5957, 533, 390, 385, 214, 5962, 5681} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, 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^3}-\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^3}-\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^3}-\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^3}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}+\frac {\log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{d^3}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 d^{5/2} \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}-\frac {b c \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 e x \left (1-c^2 x^2\right )}{8 d^2 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
[In]
[Out]
Rule 214
Rule 385
Rule 390
Rule 533
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5681
Rule 5882
Rule 5957
Rule 5959
Rule 5962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d^3 x}-\frac {e x (a+b \text {arccosh}(c x))}{d \left (d+e x^2\right )^3}-\frac {e x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^2}-\frac {e 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} \, dx}{d^3}-\frac {e \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{d^3}-\frac {e \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx}{d^2}-\frac {e \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx}{d} \\ & = \frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}-\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^3}-\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 d^2}-\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^2} \, dx}{4 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^3} \\ & = \frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{2 b d^3}-\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^3}+\frac {\sqrt {e} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^3}-\frac {\sqrt {e} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^3}-\frac {\left (b c \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 {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{2 b d^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}+\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^3}-\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^3}-\frac {\left (b c \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 {\left (b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\frac {b c \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 {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}+\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^3}+\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^3}+\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^3}-\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^3}-\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^3}-\frac {\left (b c \left (2 c^2 d+e\right ) \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 )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\frac {b c \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 {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^3}+\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^3}+\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^3}+\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^3}+\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^3} \\ & = -\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\frac {b c \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 {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^3}+\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^3}+\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^3}+\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^3}+\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^3} \\ & = -\frac {b c e x \left (1-c^2 x^2\right )}{8 d^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}+\frac {a+b \text {arccosh}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2}{b d^3}-\frac {b c \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 {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.07 (sec) , antiderivative size = 1204, normalized size of antiderivative = 1.56 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\frac {a}{4 d \left (d+e x^2\right )^2}+\frac {a}{2 d^2 \left (d+e x^2\right )}+\frac {a \log (x)}{d^3}-\frac {a \log \left (d+e x^2\right )}{2 d^3}+b \left (-\frac {5 i \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 )}{16 d^{5/2}}-\frac {5 i \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 )}{16 d^{5/2}}+\frac {\sqrt {e} \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^2}+\frac {\sqrt {e} \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^2}+\frac {\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \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 )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )}{4 d^3}-\frac {\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \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 )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )}{4 d^3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.44 (sec) , antiderivative size = 1225, normalized size of antiderivative = 1.59
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1225\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1278\) |
default | \(\text {Expression too large to display}\) | \(1278\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]
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