Optimal. Leaf size=133 \[ \frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\left (1-2 a^2\right ) b^2 \tanh ^{-1}\left (\frac {\sqrt {a+1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {b \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) x}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6321, 5468, 3785, 3919, 3831, 2659, 208} \[ \frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\left (1-2 a^2\right ) b^2 \tanh ^{-1}\left (\frac {\sqrt {a+1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {b \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 a \left (1-a^2\right ) x}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 3785
Rule 3831
Rule 3919
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \operatorname {Subst}\left (\int \frac {x \text {sech}(x) \tanh (x)}{(-a+\text {sech}(x))^3} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1-a^2-a \text {sech}(x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 a \left (1-a^2\right )}\\ &=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}+\frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {\text {sech}(x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}+\frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}+\frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a-(1+a) x^2} \, dx,x,\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}+\frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {\left (1-2 a^2\right ) b^2 \tanh ^{-1}\left (\frac {\sqrt {1+a} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [B] time = 0.97, size = 315, normalized size = 2.37 \[ \frac {1}{2} \left (-\frac {\left (2 a^2-1\right ) b^2 \log (x)}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b^2 \log (a+b x)}{a^2}+\frac {b^2 \log \left (a \sqrt {-\frac {a+b x-1}{a+b x+1}}+b x \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {-\frac {a+b x-1}{a+b x+1}}+1\right )}{a^2}+\frac {\left (2 a^2-1\right ) b^2 \log \left (\sqrt {1-a^2} a \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {1-a^2} b x \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {1-a^2} \sqrt {-\frac {a+b x-1}{a+b x+1}}-a^2-a b x+1\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {\text {sech}^{-1}(a+b x)}{x^2}-\frac {b \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1)}{(a-1) a (a+1) x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 865, normalized size = 6.50 \[ \left [-\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} + 2 \, {\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {-a^{2} + 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 2}{x^{2}}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + 2 \, {\left ({\left (a^{3} - a\right )} b^{2} x^{2} + {\left (a^{4} - a^{2}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac {2 \, {\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} - 1} b^{2} x^{2} \arctan \left (\frac {{\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {a^{2} - 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 2 \, {\left ({\left (a^{3} - a\right )} b^{2} x^{2} + {\left (a^{4} - a^{2}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\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 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 1233, normalized size = 9.27 \[ \text {result too large to display} \]
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 \[ -\frac {{\left (3 \, a^{2} b^{2} - b^{2}\right )} \log \relax (x)}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )}} + \frac {{\left (a^{4} b^{2} - 2 \, a^{3} b^{2} + a^{2} b^{2}\right )} x^{2} \log \left (b x + a + 1\right ) + {\left (a^{4} b^{2} + 2 \, a^{3} b^{2} + a^{2} b^{2}\right )} x^{2} \log \left (-b x - a + 1\right ) - 2 \, {\left (a^{3} b - a b\right )} x - 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right ) + 2 \, {\left (a^{6} - 2 \, a^{4} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (b x + a\right )}{4 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}} - \int \frac {b^{2} x + a b}{2 \, {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} - 1\right )} x^{2} + {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} - 1\right )} x^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (-b x - a + 1\right )\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asech}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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