Optimal. Leaf size=224 \[ \frac {2 b \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x} \]
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Rubi [A] time = 0.39, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6321, 5468, 4191, 3320, 2264, 2190, 2279, 2391} \[ \frac {2 b \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3320
Rule 4191
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)^2}{x^2} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x^2 \text {sech}(x) \tanh (x)}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)^2}{x}+(2 b) \operatorname {Subst}\left (\int \frac {x}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)^2}{x}+(2 b) \operatorname {Subst}\left (\int \left (-\frac {x}{a}+\frac {x}{a (1-a \cosh (x))}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {x}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^x x}{-a+2 e^x-a e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^x x}{2-2 \sqrt {1-a^2}-2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^x x}{2+2 \sqrt {1-a^2}-2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}+\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(2 b) \operatorname {Subst}\left (\int \log \left (1-\frac {2 a e^x}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}+\frac {(2 b) \operatorname {Subst}\left (\int \log \left (1-\frac {2 a e^x}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}+\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 a x}{2-2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 a x}{2+2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^2}{a}-\frac {\text {sech}^{-1}(a+b x)^2}{x}+\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [C] time = 2.69, size = 678, normalized size = 3.03 \[ \frac {-\frac {(a+b x) \text {sech}^{-1}(a+b x)^2}{x}+\frac {2 b \left (i \left (\text {Li}_2\left (\frac {\left (-i \sqrt {a^2-1}-1\right ) \left (a-i \sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )-1\right )}{a \left (a+i \sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )-1\right )}\right )-\text {Li}_2\left (\frac {\left (\sqrt {a^2-1}+i\right ) \left (a-i \sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )-1\right )}{a \left (\sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )-i (a-1)\right )}\right )\right )+2 \text {sech}^{-1}(a+b x) \tan ^{-1}\left (\frac {(a-1) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )-2 i \cos ^{-1}\left (\frac {1}{a}\right ) \tan ^{-1}\left (\frac {(a+1) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )-\log \left (-\frac {(a-1) \left (-i \sqrt {a^2-1}+a+1\right ) \left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )-1\right )}{a \left (i \sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (2 \tan ^{-1}\left (\frac {(a+1) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )+\cos ^{-1}\left (\frac {1}{a}\right )\right )-\log \left (\frac {(a-1) \left (i \sqrt {a^2-1}+a+1\right ) \left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )+1\right )}{a \left (i \sqrt {a^2-1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (\cos ^{-1}\left (\frac {1}{a}\right )-2 \tan ^{-1}\left (\frac {(a+1) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )\right )+\log \left (\frac {\sqrt {a^2-1} e^{-\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right ) \left (2 \left (\tan ^{-1}\left (\frac {(a-1) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )+\tan ^{-1}\left (\frac {(a+1) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )\right )+\cos ^{-1}\left (\frac {1}{a}\right )\right )+\log \left (\frac {\sqrt {a^2-1} e^{\frac {1}{2} \text {sech}^{-1}(a+b x)}}{\sqrt {2} \sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right ) \left (\cos ^{-1}\left (\frac {1}{a}\right )-2 \left (\tan ^{-1}\left (\frac {(a-1) \coth \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )+\tan ^{-1}\left (\frac {(a+1) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {a^2-1}}\right )\right )\right )\right )}{\sqrt {a^2-1}}}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 367, normalized size = 1.64 \[ -\frac {b \mathrm {arcsech}\left (b x +a \right )^{2}}{a}-\frac {\mathrm {arcsech}\left (b x +a \right )^{2}}{x}+\frac {2 b \sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}-\frac {2 b \sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}+\frac {2 b \sqrt {-a^{2}+1}\, \dilog \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )}-\frac {2 b \sqrt {-a^{2}+1}\, \dilog \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a \left (a^{2}-1\right )} \]
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 \[ -\frac {\log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2}}{x} - \int -\frac {2 \, {\left (2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b - b\right )} x - 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) - {\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) - {\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + {\left (2 \, a^{2} b - b\right )} x - {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )\right )}}{b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2} + {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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