Integrand size = 12, antiderivative size = 378 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(a+b x)}\right ) \]
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Time = 0.39 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6456, 5714, 5689, 3799, 2221, 2611, 6744, 2320, 6724, 5681} \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(a+b x)}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5681
Rule 5689
Rule 5714
Rule 6456
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^3 \text {sech}(x) \tanh (x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = -\text {Subst}\left (\int \frac {x^3 \tanh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = -\left (a \text {Subst}\left (\int \frac {x^3 \sinh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-\text {Subst}\left (\int x^3 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-a \text {Subst}\left (\int \frac {e^x x^3}{1-\sqrt {1-a^2}-a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^x x^3}{1+\sqrt {1-a^2}-a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1-\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-3 \text {Subst}\left (\int x^2 \log \left (1-\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+3 \text {Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )-\frac {3}{2} \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right ) \\ & = \text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a+b x)}\right )+6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )+6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right ) \\ & = \text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(a+b x)}\right ) \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.02 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=-\frac {1}{2} \text {sech}^{-1}(a+b x)^4-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1+\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{-2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{-2 \text {sech}^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (4,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right ) \]
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\[\int \frac {\operatorname {arcsech}\left (b x +a \right )^{3}}{x}d x\]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int \frac {\operatorname {asech}^{3}{\left (a + b x \right )}}{x}\, dx \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^3}{x} \,d x \]
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