Optimal. Leaf size=80 \[ \frac {2 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b}-\frac {2 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {4 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6315, 6279, 5418, 4180, 2279, 2391} \[ \frac {2 i \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b}-\frac {2 i \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {4 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 5418
Rule 6279
Rule 6315
Rubi steps
\begin {align*} \int \text {sech}^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {4 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b}+\frac {(2 i) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b}-\frac {(2 i) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {4 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{b}\\ &=\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{b}-\frac {4 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a+b x)}\right )}{b}+\frac {2 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a+b x)}\right )}{b}-\frac {2 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 105, normalized size = 1.31 \[ \frac {i \left (2 \text {Li}_2\left (-i e^{-\text {sech}^{-1}(a+b x)}\right )-2 \text {Li}_2\left (i e^{-\text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x) \left (-i (a+b x) \text {sech}^{-1}(a+b x)+2 \log \left (1-i e^{-\text {sech}^{-1}(a+b x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(a+b x)}\right )\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arsech}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int \operatorname {arsech}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 209, normalized size = 2.61 \[ x \mathrm {arcsech}\left (b x +a \right )^{2}+\frac {\mathrm {arcsech}\left (b x +a \right )^{2} a}{b}+\frac {2 i \mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b}-\frac {2 i \mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b}+\frac {2 i \dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b}-\frac {2 i \dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )}{b} \]
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 \[ x \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2} - \int -\frac {2 \, {\left (2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2} - {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b - b\right )} x + 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) + {\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) + {\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + {\left (2 \, a^{2} b - b\right )} x + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )\right )}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} + {\left (3 \, a^{2} b - b\right )} x - a}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
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 \[ \int \operatorname {asech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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