Optimal. Leaf size=62 \[ -\frac {b \log (x)}{a^2+1}+\frac {b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}+\frac {a b \tan ^{-1}(a+b x)}{a^2+1}-\frac {\cot ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.04, antiderivative size = 62, 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 = {5046, 371, 706, 31, 635, 203, 260} \[ -\frac {b \log (x)}{a^2+1}+\frac {b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}+\frac {a b \tan ^{-1}(a+b x)}{a^2+1}-\frac {\cot ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 31
Rule 203
Rule 260
Rule 371
Rule 635
Rule 706
Rule 5046
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{x^2} \, dx &=-\frac {\cot ^{-1}(a+b x)}{x}-b \int \frac {1}{x \left (1+(a+b x)^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a+b x)}{x}-b \operatorname {Subst}\left (\int \frac {1}{(-a+x) \left (1+x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {\cot ^{-1}(a+b x)}{x}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{1+a^2}-\frac {b \operatorname {Subst}\left (\int \frac {-a-x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}\\ &=-\frac {\cot ^{-1}(a+b x)}{x}-\frac {b \log (x)}{1+a^2}+\frac {b \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}+\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}\\ &=-\frac {\cot ^{-1}(a+b x)}{x}+\frac {a b \tan ^{-1}(a+b x)}{1+a^2}-\frac {b \log (x)}{1+a^2}+\frac {b \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 66, normalized size = 1.06 \[ -\frac {\cot ^{-1}(a+b x)}{x}+\frac {b ((1-i a) \log (-a-b x+i)+(1+i a) \log (a+b x+i)-2 \log (x))}{2 \left (a^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 64, normalized size = 1.03 \[ \frac {2 \, a b x \arctan \left (b x + a\right ) + b x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, b x \log \relax (x) - 2 \, {\left (a^{2} + 1\right )} \operatorname {arccot}\left (b x + a\right )}{2 \, {\left (a^{2} + 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 498, normalized size = 8.03 \[ -\frac {{\left (2 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 2 \, a \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 2 \, a \arctan \left (\frac {1}{b x + a}\right ) - 4 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - \log \left (\frac {4 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 4 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right )\right )} b}{2 \, {\left (2 \, a^{3} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - a^{2} + 2 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 63, normalized size = 1.02 \[ -\frac {\mathrm {arccot}\left (b x +a \right )}{x}-\frac {b \ln \left (b x \right )}{a^{2}+1}+\frac {b \ln \left (1+\left (b x +a \right )^{2}\right )}{2 a^{2}+2}+\frac {a b \arctan \left (b x +a \right )}{a^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 77, normalized size = 1.24 \[ \frac {1}{2} \, b {\left (\frac {2 \, a \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{2} + 1} + \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{2} + 1} - \frac {2 \, \log \relax (x)}{a^{2} + 1}\right )} - \frac {\operatorname {arccot}\left (b x + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 62, normalized size = 1.00 \[ -\frac {\mathrm {acot}\left (a+b\,x\right )}{x}-\frac {b\,x\,\ln \relax (x)-\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}+a\,b\,x\,\mathrm {acot}\left (a+b\,x\right )}{x\,\left (a^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.77, size = 167, normalized size = 2.69 \[ \begin {cases} - \frac {i b \operatorname {acot}{\left (b x - i \right )}}{2} - \frac {\operatorname {acot}{\left (b x - i \right )}}{x} + \frac {i}{2 x} & \text {for}\: a = - i \\\frac {i b \operatorname {acot}{\left (b x + i \right )}}{2} - \frac {\operatorname {acot}{\left (b x + i \right )}}{x} - \frac {i}{2 x} & \text {for}\: a = i \\- \frac {2 a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac {2 a b x \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac {2 b x \log {\relax (x )}}{2 a^{2} x + 2 x} + \frac {b x \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 a^{2} x + 2 x} - \frac {2 \operatorname {acot}{\left (a + b x \right )}}{2 a^{2} x + 2 x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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