Integrand size = 21, antiderivative size = 778 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3} \]
[Out]
Time = 1.07 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6438, 5959, 5882, 3799, 2221, 2317, 2438, 5957, 533, 390, 385, 214, 5962, 5681} \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 e^3}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (\frac {d}{x^2}+e\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\frac {d}{x^2}+e\right )^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}-\frac {\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {b \sqrt {\frac {1}{c^2 x^2}-1} \left (c^2 d+2 e\right ) \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{8 e^{5/2} \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \left (c^2 d+e\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \left (c^2 d+e\right ) \left (\frac {d}{x^2}+e\right )}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e^3} \]
[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
Rule 6438
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{x \left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{e^3 x}-\frac {d x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )^3}-\frac {d x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}-\frac {d x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{e^3 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {sech}^{-1}(c x)\right )}{b e^3}+\frac {d \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e^2}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{4 c e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b e^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 {sech}^{-1}(c x)\right )}{b e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^3}+\frac {\left (b \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {\left (b \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x^2}{c^2}} \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{4 c e \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = \frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b e^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^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 {sech}^{-1}(c x)\right )}{e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}+\frac {\left (b \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{e-\left (d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 c e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {\left (b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{8 c e^2 \left (c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = \frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {sech}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}+\frac {\left (b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{e-\left (d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 c e^2 \left (c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = \frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {sech}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 e^3} \\ & = \frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {sech}^{-1}(c x)}{b}\right )}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 e^3}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 e^3} \\ & = \frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {sech}^{-1}(c x)}{b}\right )}\right )}{2 e^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.93 (sec) , antiderivative size = 2000, normalized size of antiderivative = 2.57 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {a d^2}{4 e^3 \left (d+e x^2\right )^2}+\frac {a d}{e^3 \left (d+e x^2\right )}+\frac {a \log \left (d+e x^2\right )}{2 e^3}+b \left (-\frac {d \left (-\frac {i \sqrt {e} \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{\sqrt {d} \left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {sech}^{-1}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {\log (x)}{d \sqrt {e}}-\frac {\log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d \sqrt {e}}+\frac {\left (2 c^2 d+e\right ) \log \left (-\frac {4 d \sqrt {e} \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (2 c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (c^2 d+e\right )^{3/2}}\right )}{16 e^{5/2}}-\frac {d \left (\frac {i \sqrt {e} \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{\sqrt {d} \left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {sech}^{-1}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {\log (x)}{d \sqrt {e}}-\frac {\log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d \sqrt {e}}+\frac {\left (2 c^2 d+e\right ) \log \left (-\frac {4 d \sqrt {e} \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (2 c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (c^2 d+e\right )^{3/2}}\right )}{16 e^{5/2}}-\frac {7 i \sqrt {d} \left (-\frac {\text {sech}^{-1}(c x)}{i \sqrt {d} \sqrt {e}+e x}+\frac {i \left (\frac {\log (x)}{\sqrt {e}}-\frac {\log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{\sqrt {e}}+\frac {\log \left (\frac {2 i \sqrt {e} \left (\sqrt {d} \sqrt {\frac {1-c x}{1+c x}} (1+c x)+\frac {\sqrt {d} \sqrt {e}+i c^2 d x}{\sqrt {c^2 d+e}}\right )}{i \sqrt {d}+\sqrt {e} x}\right )}{\sqrt {c^2 d+e}}\right )}{\sqrt {d}}\right )}{16 e^{5/2}}+\frac {7 i \sqrt {d} \left (-\frac {\text {sech}^{-1}(c x)}{-i \sqrt {d} \sqrt {e}+e x}-\frac {i \left (\frac {\log (x)}{\sqrt {e}}-\frac {\log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{\sqrt {e}}+\frac {\log \left (\frac {2 \sqrt {e} \left (i \sqrt {d} \sqrt {\frac {1-c x}{1+c x}} (1+c x)+\frac {i \sqrt {d} \sqrt {e}+c^2 d x}{\sqrt {c^2 d+e}}\right )}{-i \sqrt {d}+\sqrt {e} x}\right )}{\sqrt {c^2 d+e}}\right )}{\sqrt {d}}\right )}{16 e^{5/2}}+\frac {\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )-2 \left (-4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 e^3}-\frac {-\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )+2 \left (-4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-\text {sech}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 e^3}\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.87 (sec) , antiderivative size = 1549, normalized size of antiderivative = 1.99
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1549\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1562\) |
default | \(\text {Expression too large to display}\) | \(1562\) |
[In]
[Out]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
[In]
[Out]