Optimal. Leaf size=260 \[ -\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 b^2}+\frac {6 i a \text {Li}_3\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {Li}_3\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {3 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )}{b^2}+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.26, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6321, 5468, 4190, 4180, 2531, 2282, 6589, 4184, 3718, 2190, 2279, 2391} \[ -\frac {6 i a \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 b^2}+\frac {6 i a \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}-\frac {3 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )}{b^2}+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3718
Rule 4180
Rule 4184
Rule 4190
Rule 5468
Rule 6321
Rule 6589
Rubi steps
\begin {align*} \int x \text {sech}^{-1}(a+b x)^3 \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \text {sech}(x) (-a+\text {sech}(x)) \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3-\frac {3 \operatorname {Subst}\left (\int x^2 (-a+\text {sech}(x))^2 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3-\frac {3 \operatorname {Subst}\left (\int \left (a^2 x^2-2 a x^2 \text {sech}(x)+x^2 \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3-\frac {3 \operatorname {Subst}\left (\int x^2 \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}+\frac {(3 a) \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}-\frac {(6 i a) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}+\frac {(6 i a) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}+\frac {(6 i a) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}-\frac {(6 i a) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {3 \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}+\frac {(6 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {(6 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}\\ &=-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {Li}_3\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {Li}_3\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 b^2}\\ &=-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 b^2}+\frac {6 i a \text {Li}_3\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {Li}_3\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 254, normalized size = 0.98 \[ \frac {-3 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a+b x)}\right )+6 i a \left (-2 \text {sech}^{-1}(a+b x) \left (\text {Li}_2\left (-i e^{-\text {sech}^{-1}(a+b x)}\right )-\text {Li}_2\left (i e^{-\text {sech}^{-1}(a+b x)}\right )\right )-2 \text {Li}_3\left (-i e^{-\text {sech}^{-1}(a+b x)}\right )+2 \text {Li}_3\left (i e^{-\text {sech}^{-1}(a+b x)}\right )-\left (\text {sech}^{-1}(a+b x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(a+b x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a+b x)}\right )\right )\right )\right )+(a+b x)^2 \text {sech}^{-1}(a+b x)^3-2 a (a+b x) \text {sech}^{-1}(a+b x)^3-3 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2+3 \text {sech}^{-1}(a+b x) \left (\text {sech}^{-1}(a+b x)+2 \log \left (e^{-2 \text {sech}^{-1}(a+b x)}+1\right )\right )}{2 b^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arsech}\left (b x + a\right )^{3}, x\right ) \]
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 \[ \int x \operatorname {arsech}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.88, size = 0, normalized size = 0.00 \[ \int x \mathrm {arcsech}\left (b x +a \right )^{3}\, dx \]
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 {1}{2} \, x^{2} \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} - \int \frac {16 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{3} + 16 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right )^{3} + 3 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + {\left (a^{2} b - b\right )} x^{2} + 4 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right ) + {\left (2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) + {\left (2 \, b^{3} x^{4} + 4 \, a b^{2} x^{3} + {\left (2 \, a^{2} b - b\right )} x^{2} + 2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\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^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\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^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\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} + {\left (3 \, a^{2} b - b\right )} x - a\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 \[ \int x\,{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \]
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 \[ \int x \operatorname {asech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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