Optimal. Leaf size=70 \[ \frac {2 b \tan ^{-1}\left (\frac {\sqrt {a+1} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a \sqrt {1-a^2}}-\frac {b \sec ^{-1}(a+b x)}{a}-\frac {\sec ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5258, 4426, 3783, 2659, 205} \[ \frac {2 b \tan ^{-1}\left (\frac {\sqrt {a+1} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a \sqrt {1-a^2}}-\frac {b \sec ^{-1}(a+b x)}{a}-\frac {\sec ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 3783
Rule 4426
Rule 5258
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}(a+b x)}{x^2} \, dx &=b \operatorname {Subst}\left (\int \frac {x \sec (x) \tan (x)}{(-a+\sec (x))^2} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=-\frac {\sec ^{-1}(a+b x)}{x}+b \operatorname {Subst}\left (\int \frac {1}{-a+\sec (x)} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=-\frac {b \sec ^{-1}(a+b x)}{a}-\frac {\sec ^{-1}(a+b x)}{x}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-a \cos (x)} \, dx,x,\sec ^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \sec ^{-1}(a+b x)}{a}-\frac {\sec ^{-1}(a+b x)}{x}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{1-a+(1+a) x^2} \, dx,x,\tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{a}\\ &=-\frac {b \sec ^{-1}(a+b x)}{a}-\frac {\sec ^{-1}(a+b x)}{x}+\frac {2 b \tan ^{-1}\left (\frac {\sqrt {1+a} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 112, normalized size = 1.60 \[ -\frac {\sec ^{-1}(a+b x)}{x}+\frac {b \left (\sin ^{-1}\left (\frac {1}{a+b x}\right )-\frac {i \log \left (\frac {2 \left (a \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac {i a \left (a^2+a b x-1\right )}{\sqrt {1-a^2}}\right )}{b x}\right )}{\sqrt {1-a^2}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 281, normalized size = 4.01 \[ \left [-\frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {a^{2} - 1} b x \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) + {\left (a^{3} - a\right )} \operatorname {arcsec}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}, -\frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, \sqrt {-a^{2} + 1} b x \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) + {\left (a^{3} - a\right )} \operatorname {arcsec}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 94, normalized size = 1.34 \[ b {\left (\frac {2 \, \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{\sqrt {-a^{2} + 1} a} + \frac {\arccos \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a {\left (\frac {a}{b x + a} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 154, normalized size = 2.20 \[ -\frac {\mathrm {arcsec}\left (b x +a \right )}{x}+\frac {b \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a}-\frac {b \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}} \]
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 \[ \frac {b x \int \frac {1}{\sqrt {b x + a + 1} \sqrt {b x + a - 1} b x^{2} + \sqrt {b x + a + 1} \sqrt {b x + a - 1} a x}\,{d x} - \arctan \left (\sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )}{x} \]
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 \[ \int \frac {\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asec}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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