Optimal. Leaf size=100 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}+\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}+\frac {1}{2} b c \log \left (1-\frac {1}{c^2 x^2+1}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )-\frac {1}{2} b c e \text {Li}_2\left (\frac {1}{c^2 x^2+1}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 92, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5017, 2475, 2411, 2344, 2301, 2316, 2315, 4884} \[ -\frac {1}{2} b c e \text {PolyLog}\left (2,-c^2 x^2\right )-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}+\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}-\frac {b c \left (e \log \left (c^2 x^2+1\right )+d\right )^2}{4 e}+b c d \log (x) \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2315
Rule 2316
Rule 2344
Rule 2411
Rule 2475
Rule 4884
Rule 5017
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+(b c) \int \frac {d+e \log \left (1+c^2 x^2\right )}{x \left (1+c^2 x^2\right )} \, dx+\left (2 c^2 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx\\ &=\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log \left (1+c^2 x\right )}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x \left (-\frac {1}{c^2}+\frac {x}{c^2}\right )} \, dx,x,1+c^2 x^2\right )}{2 c}\\ &=\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )}{2 c}-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x} \, dx,x,1+c^2 x^2\right )\\ &=\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}+b c d \log (x)-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{4 e}+\frac {(b e) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )}{2 c}\\ &=\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}+b c d \log (x)-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{4 e}-\frac {1}{2} b c e \text {Li}_2\left (-c^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 111, normalized size = 1.11 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}+\frac {c e \left (a+b \tan ^{-1}(c x)\right )^2}{b}+b c \left (\frac {1}{2} e \text {Li}_2\left (c^2 x^2+1\right )-\frac {\left (e \log \left (c^2 x^2+1\right )+d\right ) \left (-2 e \log \left (-c^2 x^2\right )+e \log \left (c^2 x^2+1\right )+d\right )}{4 e}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \arctan \left (c x\right ) + a d + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (c^{2} x^{2} + 1\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 11.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d + {\left (2 \, c \arctan \left (c x\right ) - \frac {\log \left (c^{2} x^{2} + 1\right )}{x}\right )} a e + b e \int \frac {\arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{x^{2}}\,{d x} - \frac {a d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 141.61, size = 160, normalized size = 1.60 \[ - \frac {a d}{x} + \frac {2 a e \operatorname {atan}{\left (\frac {x}{\sqrt {\frac {1}{c^{2}}}} \right )}}{\sqrt {\frac {1}{c^{2}}}} - \frac {a e \log {\left (c^{2} x^{2} + 1 \right )}}{x} - b c^{3} e \left (\begin {cases} 0 & \text {for}\: c^{2} = 0 \\\frac {\log {\left (c^{2} x^{2} + 1 \right )}^{2}}{4 c^{2}} & \text {otherwise} \end {cases}\right ) + 4 b c^{2} e \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\operatorname {atan}^{2}{\left (c x \right )}}{4 c} & \text {otherwise} \end {cases}\right ) - \frac {b c d \log {\left (c^{2} + \frac {1}{x^{2}} \right )}}{2} - \frac {b c e \operatorname {Li}_{2}\left (c^{2} x^{2} e^{i \pi }\right )}{2} - \frac {b d \operatorname {atan}{\left (c x \right )}}{x} - \frac {b e \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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