Integrand size = 10, antiderivative size = 203 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=-\frac {\left (2+17 a^2\right ) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^4}-\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)+\frac {a \left (1+2 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{2 b^4} \]
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Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6456, 5576, 3867, 4133, 3855, 3852, 8} \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {\left (2 a^2+1\right ) a \arctan \left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{2 b^4}-\frac {\left (17 a^2+2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^4}+\frac {a (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{3 b^4}-\frac {x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x) \]
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Rule 8
Rule 3852
Rule 3855
Rule 3867
Rule 4133
Rule 5576
Rule 6456
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x \text {sech}(x) (-a+\text {sech}(x))^3 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^4} \\ & = \frac {1}{4} x^4 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {sech}(x))^4 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{4 b^4} \\ & = -\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {sech}(x)) \left (-3 a^3+\left (2+9 a^2\right ) \text {sech}(x)-8 a \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{12 b^4} \\ & = -\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)-\frac {\text {Subst}\left (\int \left (6 a^4-12 a \left (1+2 a^2\right ) \text {sech}(x)+2 \left (2+17 a^2\right ) \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{24 b^4} \\ & = -\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)+\frac {\left (a \left (1+2 a^2\right )\right ) \text {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^4}-\frac {\left (2+17 a^2\right ) \text {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{12 b^4} \\ & = -\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)+\frac {a \left (1+2 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{2 b^4}-\frac {\left (i \left (2+17 a^2\right )\right ) \text {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)\right )}{12 b^4} \\ & = -\frac {\left (2+17 a^2\right ) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^4}-\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)+\frac {a \left (1+2 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{2 b^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.11 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=-\frac {\sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (2+2 a+13 a^2+13 a^3+\left (2-4 a+9 a^2\right ) b x+(1-3 a) b^2 x^2+b^3 x^3\right )-3 b^4 x^4 \text {sech}^{-1}(a+b x)-3 a^4 \log (a+b x)+3 a^4 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )+6 i a \left (1+2 a^2\right ) \log \left (-2 i (a+b x)+2 \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)\right )}{12 b^4} \]
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Time = 0.84 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsech}\left (b x +a \right ) a^{4}}{4}-\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+12 a^{3} \arcsin \left (b x +a \right )+18 a^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+6 a \arcsin \left (b x +a \right )+2 \sqrt {1-\left (b x +a \right )^{2}}\right )}{12 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{4}}\) | \(250\) |
default | \(\frac {\frac {\operatorname {arcsech}\left (b x +a \right ) a^{4}}{4}-\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+12 a^{3} \arcsin \left (b x +a \right )+18 a^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+6 a \arcsin \left (b x +a \right )+2 \sqrt {1-\left (b x +a \right )^{2}}\right )}{12 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{4}}\) | \(250\) |
parts | \(\frac {x^{4} \operatorname {arcsech}\left (b x +a \right )}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 \,\operatorname {csgn}\left (b \right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{4}+\operatorname {csgn}\left (b \right ) b^{2} x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-4 \,\operatorname {csgn}\left (b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +13 \,\operatorname {csgn}\left (b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+12 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{3}+2 \,\operatorname {csgn}\left (b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+6 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a \right ) \operatorname {csgn}\left (b \right )}{12 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\) | \(296\) |
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Time = 0.31 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.70 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\frac {6 \, b^{4} x^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 12 \, {\left (2 \, a^{3} + a\right )} \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} + {\left (9 \, a^{2} + 2\right )} b x + 2 \, a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{24 \, b^{4}} \]
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\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int x^{3} \operatorname {asech}{\left (a + b x \right )}\, dx \]
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\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arsech}\left (b x + a\right ) \,d x } \]
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\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arsech}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int x^3\,\mathrm {acosh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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