Optimal. Leaf size=279 \[ \frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {2 i a^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {2 i a^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {i \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 a \log (a+b x)}{b^3}+\frac {2 a \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{3 b^3}-\frac {2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {x}{3 b^2} \]
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Rubi [A] time = 0.24, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6321, 5468, 4190, 4180, 2279, 2391, 4184, 3475, 4185} \[ \frac {2 i a^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {2 i a^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {i \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {2 a \log (a+b x)}{b^3}+\frac {2 a \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{3 b^3}-\frac {2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4180
Rule 4184
Rule 4185
Rule 4190
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int x^2 \text {sech}^{-1}(a+b x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}(x) (-a+\text {sech}(x))^2 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int x (-a+\text {sech}(x))^3 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int \left (-a^3 x+3 a^2 x \text {sech}(x)-3 a x \text {sech}^2(x)+x \text {sech}^3(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int x \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \operatorname {Subst}\left (\int x \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {x}{3 b^2}+\frac {2 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}+\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {x}{3 b^2}+\frac {2 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {2 a \log (a+b x)}{b^3}+\frac {i \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}-\frac {i \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{3 b^3}+\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}\\ &=-\frac {x}{3 b^2}+\frac {2 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {2 a \log (a+b x)}{b^3}+\frac {2 i a^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {2 i a^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}\\ &=-\frac {x}{3 b^2}+\frac {2 a \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^3}-\frac {(a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{3 b^3}+\frac {a^3 \text {sech}^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a+b x)^2-\frac {2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {4 a^2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}+\frac {2 a \log (a+b x)}{b^3}+\frac {i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 305, normalized size = 1.09 \[ -\frac {-\left (6 a^2+1\right ) \left (2 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )-2 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )-2 i \text {sech}^{-1}(a+b x) \log \left (1-i e^{\text {sech}^{-1}(a+b x)}\right )+\pi \log \left (1-i e^{\text {sech}^{-1}(a+b x)}\right )+2 i \text {sech}^{-1}(a+b x) \log \left (1+i e^{\text {sech}^{-1}(a+b x)}\right )-\pi \log \left (1+i e^{\text {sech}^{-1}(a+b x)}\right )-\pi \log \left (\cot \left (\frac {1}{4} \left (\pi +2 i \text {sech}^{-1}(a+b x)\right )\right )\right )\right )+2 \left (-3 a^2 (a+b x) \text {sech}^{-1}(a+b x)^2-6 a \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)+a+b x\right )+12 a \log \left (\frac {1}{a+b x}\right )-2 (a+b x)^3 \text {sech}^{-1}(a+b x)^2+6 a (a+b x)^2 \text {sech}^{-1}(a+b x)^2+2 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1) (a+b x) \text {sech}^{-1}(a+b x)}{6 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arsech}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arsech}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.68, size = 655, normalized size = 2.35 \[ -\frac {a}{3 b^{3}}-\frac {x}{3 b^{2}}-\frac {2 a \ln \left (1+\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{b^{3}}+\frac {4 a \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{b^{3}}+\frac {2 i a^{2} \dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{3}}+\frac {a^{3} \mathrm {arcsech}\left (b x +a \right )^{2}}{3 b^{3}}+\frac {4 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, \mathrm {arcsech}\left (b x +a \right ) x a}{3 b^{2}}-\frac {2 a \,\mathrm {arcsech}\left (b x +a \right )}{b^{3}}-\frac {2 i a^{2} \dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{3}}-\frac {i \mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{3 b^{3}}+\frac {2 i a^{2} \mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{3}}+\frac {x^{3} \mathrm {arcsech}\left (b x +a \right )^{2}}{3}-\frac {2 i a^{2} \mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{3}}+\frac {i \dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{3 b^{3}}+\frac {5 \sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, \mathrm {arcsech}\left (b x +a \right ) a^{2}}{3 b^{3}}-\frac {i \dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{3 b^{3}}+\frac {i \mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{3 b^{3}}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, \mathrm {arcsech}\left (b x +a \right ) x^{2}}{3 b} \]
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 \[ \frac {1}{3} \, x^{3} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2} - \int -\frac {2 \, {\left (6 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + 6 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )^{2} - {\left (b^{3} x^{5} + 2 \, a b^{2} x^{4} + {\left (a^{2} b - b\right )} x^{3} + 6 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right ) + {\left (3 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) + {\left (2 \, b^{3} x^{5} + 4 \, a b^{2} x^{4} + {\left (2 \, a^{2} b - b\right )} x^{3} + 3 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )\right )}}{3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} + {\left (3 \, a^{2} b - b\right )} x - a\right )}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {asech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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