Optimal. Leaf size=62 \[ -\frac {a b \text {ArcTan}(a+b x)}{1+a^2}-\frac {\text {ArcTan}(a+b x)}{x}+\frac {b \log (x)}{1+a^2}-\frac {b \log \left (1+(a+b x)^2\right )}{2 \left (1+a^2\right )} \]
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Rubi [A]
time = 0.03, 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 = {5153, 378, 720,
31, 649, 209, 266} \begin {gather*} -\frac {a b \text {ArcTan}(a+b x)}{a^2+1}+\frac {b \log (x)}{a^2+1}-\frac {b \log \left ((a+b x)^2+1\right )}{2 \left (a^2+1\right )}-\frac {\text {ArcTan}(a+b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 209
Rule 266
Rule 378
Rule 649
Rule 720
Rule 5153
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{x^2} \, dx &=-\frac {\tan ^{-1}(a+b x)}{x}+b \int \frac {1}{x \left (1+(a+b x)^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(a+b x)}{x}+b \text {Subst}\left (\int \frac {1}{(-a+x) \left (1+x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {\tan ^{-1}(a+b x)}{x}+\frac {b \text {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{1+a^2}+\frac {b \text {Subst}\left (\int \frac {-a-x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}\\ &=-\frac {\tan ^{-1}(a+b x)}{x}+\frac {b \log (x)}{1+a^2}-\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}-\frac {(a b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{1+a^2}\\ &=-\frac {a b \tan ^{-1}(a+b x)}{1+a^2}-\frac {\tan ^{-1}(a+b x)}{x}+\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] Result contains complex when optimal does not.
time = 0.04, size = 67, normalized size = 1.08 \begin {gather*} -\frac {\text {ArcTan}(a+b x)}{x}+\frac {b (2 \log (x)+i (i+a) \log (i-a-b x)+(-1-i a) \log (i+a+b x))}{2 \left (1+a^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 61, normalized size = 0.98
method | result | size |
derivativedivides | \(b \left (-\frac {\arctan \left (b x +a \right )}{b x}-\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+\arctan \left (b x +a \right ) a}{a^{2}+1}+\frac {\ln \left (-b x \right )}{a^{2}+1}\right )\) | \(61\) |
default | \(b \left (-\frac {\arctan \left (b x +a \right )}{b x}-\frac {\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}+\arctan \left (b x +a \right ) a}{a^{2}+1}+\frac {\ln \left (-b x \right )}{a^{2}+1}\right )\) | \(61\) |
risch | \(\frac {i \ln \left (1+i \left (b x +a \right )\right )}{2 x}-\frac {i \left (a^{2} \ln \left (1-i \left (b x +a \right )\right )+\ln \left (1-i \left (b x +a \right )\right )-i \ln \left (\left (-a^{2} b +3 i a b \right ) x -3 a +2 i a^{2}-a^{3}\right ) b x +\ln \left (\left (-a^{2} b +3 i a b \right ) x -3 a +2 i a^{2}-a^{3}\right ) a b x -i \ln \left (\left (-a^{2} b -3 i a b \right ) x -3 a -2 i a^{2}-a^{3}\right ) b x -\ln \left (\left (-a^{2} b -3 i a b \right ) x -3 a -2 i a^{2}-a^{3}\right ) a b x +2 i \ln \left (x \right ) b x \right )}{2 x \left (i+a \right ) \left (a -i\right )}\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 77, normalized size = 1.24 \begin {gather*} -\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 \left (x\right )}{a^{2} + 1}\right )} - \frac {\arctan \left (b x + a\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.10, size = 57, normalized size = 0.92 \begin {gather*} -\frac {b x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, b x \log \left (x\right ) + 2 \, {\left (a b x + a^{2} + 1\right )} \arctan \left (b x + a\right )}{2 \, {\left (a^{2} + 1\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 168, normalized size = 2.71 \begin {gather*} \begin {cases} - \frac {i b \operatorname {atan}{\left (b x - i \right )}}{2} - \frac {\operatorname {atan}{\left (b x - i \right )}}{x} - \frac {i}{2 x} & \text {for}\: a = - i \\\frac {i b \operatorname {atan}{\left (b x + i \right )}}{2} - \frac {\operatorname {atan}{\left (b x + i \right )}}{x} + \frac {i}{2 x} & \text {for}\: a = i \\- \frac {2 a^{2} \operatorname {atan}{\left (a + b x \right )}}{2 a^{2} x + 2 x} - \frac {2 a b x \operatorname {atan}{\left (a + b x \right )}}{2 a^{2} x + 2 x} + \frac {2 b x \log {\left (x \right )}}{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 {atan}{\left (a + b x \right )}}{2 a^{2} x + 2 x} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 63, normalized size = 1.02 \begin {gather*} -\frac {\mathrm {atan}\left (a+b\,x\right )}{x}-\frac {\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2}-b\,x\,\ln \left (x\right )+a\,b\,x\,\mathrm {atan}\left (a+b\,x\right )}{x\,\left (a^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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