Optimal. Leaf size=33 \[ \frac {(a+b x) \text {ArcTan}(a+b x)}{b}-\frac {\log \left (1+(a+b x)^2\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5147, 4930, 266}
\begin {gather*} \frac {(a+b x) \text {ArcTan}(a+b x)}{b}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 4930
Rule 5147
Rubi steps
\begin {align*} \int \tan ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \tan ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\log \left (1+(a+b x)^2\right )}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 39, normalized size = 1.18 \begin {gather*} -\frac {-2 (a+b x) \text {ArcTan}(a+b x)+\log \left (1+a^2+2 a b x+b^2 x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 30, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \arctan \left (b x +a \right )-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}}{b}\) | \(30\) |
default | \(\frac {\left (b x +a \right ) \arctan \left (b x +a \right )-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2}}{b}\) | \(30\) |
risch | \(-\frac {i x \ln \left (1+i \left (b x +a \right )\right )}{2}+\frac {i x \ln \left (1-i \left (b x +a \right )\right )}{2}+\frac {a \arctan \left (b x +a \right )}{b}-\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 31, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.45, size = 39, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.14, size = 46, normalized size = 1.39 \begin {gather*} \begin {cases} \frac {a \operatorname {atan}{\left (a + b x \right )}}{b} + x \operatorname {atan}{\left (a + b x \right )} - \frac {\log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {atan}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 31, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.45, size = 42, normalized size = 1.27 \begin {gather*} x\,\mathrm {atan}\left (a+b\,x\right )-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )-2\,a\,\mathrm {atan}\left (a+b\,x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________