Optimal. Leaf size=61 \[ -\frac {a+b \text {ArcTan}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left (1+(c+d x)^2\right )}{2 d e^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5151, 12, 4946,
272, 36, 29, 31} \begin {gather*} -\frac {a+b \text {ArcTan}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left ((c+d x)^2+1\right )}{2 d e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 272
Rule 4946
Rule 5151
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x \left (1+x^2\right )} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,(c+d x)^2\right )}{2 d e^2}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^2\right )}{2 d e^2}-\frac {b \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,(c+d x)^2\right )}{2 d e^2}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \log (c+d x)}{d e^2}-\frac {b \log \left (1+(c+d x)^2\right )}{2 d e^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.82 \begin {gather*} \frac {-\frac {a+b \text {ArcTan}(c+d x)}{c+d x}+b \log (c+d x)-\frac {1}{2} b \log \left (1+(c+d x)^2\right )}{d e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 65, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}-\frac {b \arctan \left (d x +c \right )}{e^{2} \left (d x +c \right )}-\frac {b \ln \left (1+\left (d x +c \right )^{2}\right )}{2 e^{2}}+\frac {b \ln \left (d x +c \right )}{e^{2}}}{d}\) | \(65\) |
default | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}-\frac {b \arctan \left (d x +c \right )}{e^{2} \left (d x +c \right )}-\frac {b \ln \left (1+\left (d x +c \right )^{2}\right )}{2 e^{2}}+\frac {b \ln \left (d x +c \right )}{e^{2}}}{d}\) | \(65\) |
risch | \(\frac {i b \ln \left (1+i \left (d x +c \right )\right )}{2 d \,e^{2} \left (d x +c \right )}-\frac {-2 \ln \left (-d x -c \right ) b d x +\ln \left (-d^{2} x^{2}-2 c d x -c^{2}-1\right ) b d x +i b \ln \left (1-i \left (d x +c \right )\right )-2 \ln \left (-d x -c \right ) b c +\ln \left (-d^{2} x^{2}-2 c d x -c^{2}-1\right ) b c +2 a}{2 e^{2} \left (d x +c \right ) d}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 86, normalized size = 1.41 \begin {gather*} -\frac {1}{2} \, {\left (d {\left (\frac {e^{\left (-2\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{2}} - \frac {2 \, e^{\left (-2\right )} \log \left (d x + c\right )}{d^{2}}\right )} + \frac {2 \, \arctan \left (d x + c\right )}{d^{2} x e^{2} + c d e^{2}}\right )} b - \frac {a}{d^{2} x e^{2} + c d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.57, size = 71, normalized size = 1.16 \begin {gather*} -\frac {{\left (2 \, b \arctan \left (d x + c\right ) + {\left (b d x + b c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 2 \, {\left (b d x + b c\right )} \log \left (d x + c\right ) + 2 \, a\right )} e^{\left (-2\right )}}{2 \, {\left (d^{2} x + c d\right )}} \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 = 2.27, size = 238, normalized size = 3.90 \begin {gather*} \begin {cases} \frac {\tilde {\infty } a}{e^{2} x} & \text {for}\: c = 0 \wedge d = 0 \\\frac {x \left (a + b \operatorname {atan}{\left (c \right )}\right )}{c^{2} e^{2}} & \text {for}\: d = 0 \\\tilde {\infty } a x & \text {for}\: c = - d x \\- \frac {a}{c d e^{2} + d^{2} e^{2} x} + \frac {b c \log {\left (\frac {c}{d} + x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b c \log {\left (\frac {c}{d} + x - \frac {i}{d} \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {i b c \operatorname {atan}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {b d x \log {\left (\frac {c}{d} + x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b d x \log {\left (\frac {c}{d} + x - \frac {i}{d} \right )}}{c d e^{2} + d^{2} e^{2} x} + \frac {i b d x \operatorname {atan}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{2} x} - \frac {b \operatorname {atan}{\left (c + d x \right )}}{c d e^{2} + d^{2} e^{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 = 0.67, size = 88, normalized size = 1.44 \begin {gather*} \frac {b\,\ln \left (c+d\,x\right )}{d\,e^2}-\frac {b\,\mathrm {atan}\left (c+d\,x\right )}{x\,d^2\,e^2+c\,d\,e^2}-\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{2\,d\,e^2}-\frac {a}{x\,d^2\,e^2+c\,d\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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