Optimal. Leaf size=57 \[ -\frac {d (a+b \text {ArcTan}(c x))}{x}+e x (a+b \text {ArcTan}(c x))+b c d \log (x)-\frac {b \left (c^2 d+e\right ) \log \left (1+c^2 x^2\right )}{2 c} \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 5096, 457,
78} \begin {gather*} -\frac {d (a+b \text {ArcTan}(c x))}{x}+e x (a+b \text {ArcTan}(c x))-\frac {b \left (c^2 d+e\right ) \log \left (c^2 x^2+1\right )}{2 c}+b c d \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 78
Rule 457
Rule 5096
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{x}+e x \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {-d+e x^2}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{x}+e x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {-d+e x}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{x}+e x \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \left (-\frac {d}{x}+\frac {c^2 d+e}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{x}+e x \left (a+b \tan ^{-1}(c x)\right )+b c d \log (x)-\frac {b \left (c^2 d+e\right ) \log \left (1+c^2 x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 73, normalized size = 1.28 \begin {gather*} -\frac {a d}{x}+a e x-\frac {b d \text {ArcTan}(c x)}{x}+b e x \text {ArcTan}(c x)+b c d \log (x)-\frac {1}{2} b c d \log \left (1+c^2 x^2\right )-\frac {b e \log \left (1+c^2 x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 84, normalized size = 1.47
method | result | size |
derivativedivides | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \arctan \left (c x \right ) e x}{c}-\frac {b \arctan \left (c x \right ) d}{c x}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d}{2}-\frac {b \ln \left (c^{2} x^{2}+1\right ) e}{2 c^{2}}+b d \ln \left (c x \right )\right )\) | \(84\) |
default | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \arctan \left (c x \right ) e x}{c}-\frac {b \arctan \left (c x \right ) d}{c x}-\frac {b \ln \left (c^{2} x^{2}+1\right ) d}{2}-\frac {b \ln \left (c^{2} x^{2}+1\right ) e}{2 c^{2}}+b d \ln \left (c x \right )\right )\) | \(84\) |
risch | \(\frac {i b \left (-e \,x^{2}+d \right ) \ln \left (i c x +1\right )}{2 x}+\frac {i b c e \,x^{2} \ln \left (-i c x +1\right )+2 b \,c^{2} d \ln \left (x \right ) x -\ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d x -i b c d \ln \left (-i c x +1\right )+2 a e \,x^{2} c -\ln \left (c^{2} x^{2}+1\right ) b e x -2 a d c}{2 c x}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 75, normalized size = 1.32 \begin {gather*} -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d + a x e + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b e}{2 \, c} - \frac {a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.05, size = 78, normalized size = 1.37 \begin {gather*} \frac {2 \, b c^{2} d x \log \left (x\right ) + 2 \, a c x^{2} e - 2 \, a c d + 2 \, {\left (b c x^{2} e - b c d\right )} \arctan \left (c x\right ) - {\left (b c^{2} d x + b x e\right )} \log \left (c^{2} x^{2} + 1\right )}{2 \, c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 80, normalized size = 1.40 \begin {gather*} \begin {cases} - \frac {a d}{x} + a e x + b c d \log {\left (x \right )} - \frac {b c d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d \operatorname {atan}{\left (c x \right )}}{x} + b e x \operatorname {atan}{\left (c x \right )} - \frac {b e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} & \text {for}\: c \neq 0 \\a \left (- \frac {d}{x} + e x\right ) & \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 = 0.23, size = 69, normalized size = 1.21 \begin {gather*} a\,e\,x-\frac {a\,d}{x}+b\,e\,x\,\mathrm {atan}\left (c\,x\right )-\frac {b\,c\,d\,\ln \left (c^2\,x^2+1\right )}{2}+b\,c\,d\,\ln \left (x\right )-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{x}-\frac {b\,e\,\ln \left (c^2\,x^2+1\right )}{2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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