Optimal. Leaf size=149 \[ -\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {2 i a \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {2 i a \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {\log (a+b x)}{b^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^2}+\frac {4 a \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2 \]
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Rubi [A] time = 0.14, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6321, 5468, 4190, 4180, 2279, 2391, 4184, 3475} \[ -\frac {2 i a \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {2 i a \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {\log (a+b x)}{b^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^2}+\frac {4 a \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4180
Rule 4184
Rule 4190
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int x \text {sech}^{-1}(a+b x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \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)^2-\frac {\operatorname {Subst}\left (\int x (-a+\text {sech}(x))^2 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \left (a^2 x-2 a x \text {sech}(x)+x \text {sech}^2(x)\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int x \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}+\frac {(2 a) \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2+\frac {4 a \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}-\frac {(2 i a) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}+\frac {(2 i a) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2+\frac {4 a \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {\log (a+b x)}{b^2}-\frac {(2 i a) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {(2 i a) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}\\ &=-\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^2+\frac {4 a \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {\log (a+b x)}{b^2}-\frac {2 i a \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {2 i a \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 172, normalized size = 1.15 \[ \frac {-4 i a \left (\text {Li}_2\left (-i e^{-\text {sech}^{-1}(a+b x)}\right )-\text {Li}_2\left (i e^{-\text {sech}^{-1}(a+b x)}\right )\right )+2 \log \left (\frac {1}{a+b x}\right )+(a+b x)^2 \text {sech}^{-1}(a+b x)^2-2 a (a+b x) \text {sech}^{-1}(a+b x)^2-2 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)-4 i a \text {sech}^{-1}(a+b x) \left (\log \left (1-i e^{-\text {sech}^{-1}(a+b x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a+b x)}\right )\right )}{2 b^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arsech}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int x \operatorname {arsech}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.31, size = 396, normalized size = 2.66 \[ \frac {x^{2} \mathrm {arcsech}\left (b x +a \right )^{2}}{2}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, \mathrm {arcsech}\left (b x +a \right ) x}{b}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \sqrt {\frac {b x +a +1}{b x +a}}\, \mathrm {arcsech}\left (b x +a \right ) a}{b^{2}}-\frac {a^{2} \mathrm {arcsech}\left (b x +a \right )^{2}}{2 b^{2}}+\frac {\mathrm {arcsech}\left (b x +a \right )}{b^{2}}-\frac {2 \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{b^{2}}+\frac {\ln \left (1+\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{b^{2}}-\frac {2 i a \,\mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{2}}+\frac {2 i a \,\mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{2}}-\frac {2 i a \dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{2}}+\frac {2 i a \dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b^{2}} \]
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 \[ \frac {1}{2} \, x^{2} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2} - \int -\frac {4 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + 4 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + {\left (a^{2} b - b\right )} x^{2} + 4 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right ) + {\left (2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) + {\left (2 \, b^{3} x^{4} + 4 \, a b^{2} x^{3} + {\left (2 \, a^{2} b - b\right )} x^{2} + 2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b - b\right )} x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} + {\left (3 \, a^{2} b - b\right )} x - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
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 \[ \int x \operatorname {asech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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