Integrand size = 21, antiderivative size = 786 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
[Out]
Time = 1.02 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6438, 5909, 5963, 95, 211, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {\frac {1}{c x}+1} \sqrt {c d-\sqrt {-d} \sqrt {e}}}{\sqrt {\frac {1}{c x}-1} \sqrt {c d+\sqrt {-d} \sqrt {e}}}\right )}{2 e \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b \arctan \left (\frac {\sqrt {\frac {1}{c x}+1} \sqrt {c d+\sqrt {-d} \sqrt {e}}}{\sqrt {\frac {1}{c x}-1} \sqrt {c d-\sqrt {-d} \sqrt {e}}}\right )}{2 e \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
[In]
[Out]
Rule 95
Rule 211
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5909
Rule 5962
Rule 5963
Rule 6438
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {d \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {d \text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}+\frac {d \text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}+\frac {d \text {Subst}\left (\int \frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{2 e} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}-d x\right )} \, dx,x,\frac {1}{x}\right )}{4 c e}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}+d x\right )} \, dx,x,\frac {1}{x}\right )}{4 c e}+\frac {d \text {Subst}\left (\int \left (-\frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \text {arccosh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{2 e} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {\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 )}{4 e^{3/2}}-\frac {\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 )}{4 e^{3/2}}-\frac {b \text {Subst}\left (\int \frac {1}{d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (-d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{2 c e}+\frac {b \text {Subst}\left (\int \frac {1}{-d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{2 c e} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {\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 )}{4 e^{3/2}}-\frac {\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 )}{4 e^{3/2}} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {\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 )}{4 e^{3/2}}-\frac {\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 )}{4 e^{3/2}}-\frac {\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 )}{4 e^{3/2}}-\frac {\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 )}{4 e^{3/2}} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}} \\ & = \frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 1226, normalized size of antiderivative = 1.56 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-\frac {2 a \sqrt {e} x}{d+e x^2}+\frac {b \text {sech}^{-1}(c x)}{i \sqrt {d}-\sqrt {e} x}-\frac {b \text {sech}^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {2 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {4 b \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 )}{\sqrt {d}}+\frac {4 b \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 )}{\sqrt {d}}-\frac {i b \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 )}{\sqrt {d}}-\frac {2 b \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 )}{\sqrt {d}}+\frac {i b \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 )}{\sqrt {d}}+\frac {2 b \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 )}{\sqrt {d}}+\frac {i b \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 )}{\sqrt {d}}-\frac {2 b \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 )}{\sqrt {d}}-\frac {i b \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 )}{\sqrt {d}}+\frac {2 b \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 )}{\sqrt {d}}+\frac {i b \sqrt {e} \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 {d} \sqrt {c^2 d+e}}-\frac {i b \sqrt {e} \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 {d} \sqrt {c^2 d+e}}-\frac {i b \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 )}{\sqrt {d}}+\frac {i b \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 )}{\sqrt {d}}+\frac {i b \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 )}{\sqrt {d}}-\frac {i b \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 )}{\sqrt {d}}}{4 e^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 46.70 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.16
method | result | size |
parts | \(-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+\frac {b \left (-\frac {c^{5} \operatorname {arcsech}\left (c x \right ) x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {c^{4} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c \,d^{3} e}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) d^{3} c}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c \,d^{3} e}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) d^{3} c}+\frac {c^{4} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{4 e}\right )}{c^{3}}\) | \(910\) |
derivativedivides | \(\frac {-\frac {a \,c^{5} x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}}{4 e}\right )}{c^{3}}\) | \(919\) |
default | \(\frac {-\frac {a \,c^{5} x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}}{4 e}\right )}{c^{3}}\) | \(919\) |
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\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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