Optimal. Leaf size=965 \[ \frac {b^2 \text {sech}^{-1}(a+b x)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}-\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)^2}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)^2}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{2 a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}-\frac {6 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac {6 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )}+\frac {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \text {Li}_3\left (\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 {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 1.34, antiderivative size = 965, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6321, 5468, 4191, 3324, 3320, 2264, 2190, 2531, 2282, 6589, 5562, 2279, 2391} \[ \frac {b^2 \text {sech}^{-1}(a+b x)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}-\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)^2}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)^2}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{2 a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}-\frac {6 b^2 \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac {6 b^2 \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right ) \text {sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {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 )}+\frac {3 b^2 \text {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 )}+\frac {6 b^2 \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \text {PolyLog}\left (3,\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 {6 b^2 \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3320
Rule 3324
Rule 4191
Rule 5468
Rule 5562
Rule 6321
Rule 6589
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx &=-\left (b^2 \operatorname {Subst}\left (\int \frac {x^3 \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)^3}{2 x^2}+\frac {1}{2} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {1}{2} \left (3 b^2\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{a^2}+\frac {x^2}{a^2 (-1+a \cosh (x))^2}+\frac {2 x^2}{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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{(-1+a \cosh (x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 a^2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2}\\ &=\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{a-2 e^x+a e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x \sinh (x)}{-1+a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}\\ &=-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{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 (3 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x}{-1-\sqrt {1-a^2}+a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x}{-1+\sqrt {1-a^2}+a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{-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 (6 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{-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 {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{-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 (3 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x^2}{-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 (3 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^x}{-1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^x}{-1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int x \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 (6 b^2\right ) \operatorname {Subst}\left (\int x \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 {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {3 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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int x \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 (3 b^2\right ) \operatorname {Subst}\left (\int x \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 (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\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 (6 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\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 {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {3 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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\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 (3 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\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 (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \sqrt {1-a^2}}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a^2 \sqrt {1-a^2}}\\ &=-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {3 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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1-\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 {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1+\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 {3 b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac {3 b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^3}{2 x^2}+\frac {3 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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac {3 b^2 \text {sech}^{-1}(a+b x)^2 \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 {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\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 {6 b^2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {3 b^2 \text {Li}_3\left (\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 {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {3 b^2 \text {Li}_3\left (\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 {6 b^2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [F] time = 9.95, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.67, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arcsech}\left (b x +a \right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{3}}{2 \, x^{2}} - \int \frac {16 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{3} + 16 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{3} - 3 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b - b\right )} x - 4 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) - {\left (2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) - {\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + {\left (2 \, a^{2} b - b\right )} x - 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2} - 24 \, {\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )}{2 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} + {\left (3 \, a^{2} b - b\right )} x^{4} + {\left (a^{3} - a\right )} x^{3} + {\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} + {\left (3 \, a^{2} b - b\right )} x^{4} + {\left (a^{3} - a\right )} x^{3}\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}^{3}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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