Optimal. Leaf size=203 \[ -\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}-\frac {\left (17 a^2+2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^4}+\frac {\left (2 a^2+1\right ) a \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{2 b^4}+\frac {a (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{3 b^4}-\frac {x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x) \]
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Rubi [A] time = 0.16, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6321, 5468, 3782, 4048, 3770, 3767, 8} \[ -\frac {\left (17 a^2+2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^4}+\frac {\left (2 a^2+1\right ) a \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{2 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}-\frac {x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{3 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4048
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int x^3 \text {sech}^{-1}(a+b x) \, dx &=-\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tan ^{-1}\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 ) \operatorname {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 ) \tan ^{-1}\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*}
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Mathematica [C] time = 0.54, size = 225, normalized size = 1.11 \[ -\frac {-3 a^4 \log (a+b x)+3 a^4 \log \left (a \sqrt {-\frac {a+b x-1}{a+b x+1}}+b x \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {-\frac {a+b x-1}{a+b x+1}}+1\right )+6 i \left (2 a^2+1\right ) a \log \left (2 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1)-2 i (a+b x)\right )+\sqrt {-\frac {a+b x-1}{a+b x+1}} \left (13 a^3+\left (9 a^2-4 a+2\right ) b x+13 a^2+(1-3 a) b^2 x^2+2 a+b^3 x^3+2\right )-3 b^4 x^4 \text {sech}^{-1}(a+b x)}{12 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 345, normalized size = 1.70 \[ \frac {6 \, b^{4} x^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 12 \, {\left (2 \, a^{3} + a\right )} \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} + {\left (9 \, a^{2} + 2\right )} b x + 2 \, a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{24 \, b^{4}} \]
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 \[ \int x^{3} \operatorname {arsech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 250, normalized size = 1.23 \[ \frac {\frac {\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right )^{3} a +\frac {3 \,\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2} a^{2}}{2}-\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right ) a^{3}+\frac {\mathrm {arcsech}\left (b x +a \right ) a^{4}}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 a^{4} \arctanh \left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+12 a^{3} \arcsin \left (b x +a \right )+\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+18 a^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 a \arcsin \left (b x +a \right )+2 \sqrt {1-\left (b x +a \right )^{2}}\right )}{12 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{4}} \]
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 \[ \frac {2 \, b^{4} x^{4} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right ) - 2 \, b^{4} x^{4} \log \left (b x + a\right ) - b^{2} x^{2} + 6 \, a b x - {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) - 2 \, {\left (b^{4} x^{4} - a^{4}\right )} \log \left (b x + a\right ) - {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (-b x - a + 1\right )}{8 \, b^{4}} + \int \frac {b^{2} x^{5} + a b x^{4}}{4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (-b x - a + 1\right )\right )} - 1\right )}}\,{d x} \]
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 \[ \int x^3\,\mathrm {acosh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x^{3} \operatorname {asech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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