Integrand size = 12, antiderivative size = 537 \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}} \]
[Out]
Time = 0.58 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6456, 5576, 4276, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=-\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}+\frac {b^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2} \]
[In]
[Out]
Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3401
Rule 3405
Rule 4276
Rule 5576
Rule 6456
Rubi steps \begin{align*} \text {integral}& = -\left (b^2 \text {Subst}\left (\int \frac {x^2 \text {sech}(x) \tanh (x)}{(-a+\text {sech}(x))^3} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right ) \\ & = -\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+b^2 \text {Subst}\left (\int \frac {x}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = -\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+b^2 \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {x}{a^2 (-1+a \cosh (x))^2}+\frac {2 x}{a^2 (-1+a \cosh (x))}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {Subst}\left (\int \frac {x}{(-1+a \cosh (x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{a-2 e^x+a e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2}-\frac {b^2 \text {Subst}\left (\int \frac {x}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \text {Subst}\left (\int \frac {\sinh (x)}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {a}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{a-2 e^x+a e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{-2-2 \sqrt {1-a^2}+2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{-2+2 \sqrt {1-a^2}+2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{-2-2 \sqrt {1-a^2}+2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )^{3/2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {e^x x}{-2+2 \sqrt {1-a^2}+2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )^{3/2}}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^x}{-2-2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \sqrt {1-a^2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^x}{-2+2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \sqrt {1-a^2}} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \text {Subst}\left (\int \log \left (1+\frac {2 a e^x}{-2-2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b^2 \text {Subst}\left (\int \log \left (1+\frac {2 a e^x}{-2+2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{-2-2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \sqrt {1-a^2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{-2+2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \sqrt {1-a^2}} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{-2-2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{-2+2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )^{3/2}} \\ & = \frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.00 (sec) , antiderivative size = 1439, normalized size of antiderivative = 2.68 \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=-\frac {(a+b x)^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 x^2}+\frac {b \text {sech}^{-1}(a+b x) \left (-a \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)+\left (-1+a^2\right ) (a+b x) \text {sech}^{-1}(a+b x)\right )}{(-1+a) a^2 (1+a) x}+\frac {b^2 \log \left (\frac {b x}{a+b x}\right )}{a^2-a^4}-\frac {2 b^2 \left (2 \text {sech}^{-1}(a+b x) \arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )-2 i \arccos \left (\frac {1}{a}\right ) \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 \left (\arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{-\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )+\left (\arccos \left (\frac {1}{a}\right )-2 \left (\arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )-\left (\arccos \left (\frac {1}{a}\right )+2 \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (-\frac {(-1+a) \left (1+a-i \sqrt {-1+a^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )-\left (\arccos \left (\frac {1}{a}\right )-2 \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (1+a+i \sqrt {-1+a^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (-1-i \sqrt {-1+a^2}\right ) \left (-1+a-i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (i+\sqrt {-1+a^2}\right ) \left (-1+a-i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-i (-1+a)+\sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )\right )\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \left (2 \text {sech}^{-1}(a+b x) \arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )-2 i \arccos \left (\frac {1}{a}\right ) \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 \left (\arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{-\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )+\left (\arccos \left (\frac {1}{a}\right )-2 \left (\arctan \left (\frac {(-1+a) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )+\arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-1+a^2} e^{\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )-\left (\arccos \left (\frac {1}{a}\right )+2 \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (-\frac {(-1+a) \left (1+a-i \sqrt {-1+a^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )-\left (\arccos \left (\frac {1}{a}\right )-2 \arctan \left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (1+a+i \sqrt {-1+a^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (-1-i \sqrt {-1+a^2}\right ) \left (-1+a-i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-1+a+i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (i+\sqrt {-1+a^2}\right ) \left (-1+a-i \sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a \left (-i (-1+a)+\sqrt {-1+a^2} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}\right )\right )\right )}{a^2 \left (-1+a^2\right )^{3/2}} \]
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Time = 1.21 (sec) , antiderivative size = 982, normalized size of antiderivative = 1.83
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right ) \left (2 \,\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}-2 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, a^{2} \left (b x +a \right )+2 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, a \left (b x +a \right )^{2}-2 \,\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}+2 a^{2}-4 \left (b x +a \right ) a +2 \left (b x +a \right )^{2}\right )}{2 a^{2} \left (a^{2}-1\right ) b^{2} x^{2}}+\frac {2 \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}-\frac {\ln \left (a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}+a -\frac {2}{b x +a}-2 \sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}\right )\) | \(982\) |
default | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right ) \left (2 \,\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}-2 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, a^{2} \left (b x +a \right )+2 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, a \left (b x +a \right )^{2}-2 \,\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}+2 a^{2}-4 \left (b x +a \right ) a +2 \left (b x +a \right )^{2}\right )}{2 a^{2} \left (a^{2}-1\right ) b^{2} x^{2}}+\frac {2 \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}-\frac {\ln \left (a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}+a -\frac {2}{b x +a}-2 \sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}\right )\) | \(982\) |
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\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {asech}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2}{x^3} \,d x \]
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