Optimal. Leaf size=189 \[ -\frac {2 c^2 e (a+b \text {ArcTan}(c x))}{3 x}-\frac {c^3 e (a+b \text {ArcTan}(c x))^2}{3 b}+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \text {ArcTan}(c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )+\frac {1}{6} b c^3 e \text {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 189, normalized size of antiderivative = 1.00, number
of steps used = 15, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules
used = {5137, 2525, 2458, 2389, 2379, 2438, 2351, 31, 5038, 4946, 272, 36, 29, 5004}
\begin {gather*} -\frac {c^3 e (a+b \text {ArcTan}(c x))^2}{3 b}-\frac {(a+b \text {ArcTan}(c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}-\frac {2 c^2 e (a+b \text {ArcTan}(c x))}{3 x}+b c^3 e \log (x)-\frac {b c \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 x^2}-\frac {1}{6} b c^3 \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}{6} b c^3 e \text {Li}_2\left (\frac {1}{c^2 x^2+1}\right )-\frac {1}{3} b c^3 e \log \left (c^2 x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rule 2525
Rule 4946
Rule 5004
Rule 5038
Rule 5137
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^4} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {1}{3} (b c) \int \frac {d+e \log \left (1+c^2 x^2\right )}{x^3 \left (1+c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {d+e \log \left (1+c^2 x\right )}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\frac {1}{3} \left (2 c^4 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx\\ &=-\frac {2 c^2 e \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \tan ^{-1}(c x)\right )^2}{3 b}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (-\frac {1}{c^2}+\frac {x}{c^2}\right )^2} \, dx,x,1+c^2 x^2\right )}{6 c}+\frac {1}{3} \left (2 b c^3 e\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {2 c^2 e \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \tan ^{-1}(c x)\right )^2}{3 b}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{\left (-\frac {1}{c^2}+\frac {x}{c^2}\right )^2} \, dx,x,1+c^2 x^2\right )}{6 c}-\frac {1}{6} (b c) \text {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 )+\frac {1}{3} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {2 c^2 e \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \tan ^{-1}(c x)\right )^2}{3 b}-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )+\frac {1}{6} \left (b c^3\right ) \text {Subst}\left (\int \frac {d+e \log (x)}{x} \, dx,x,1+c^2 x^2\right )+\frac {1}{6} (b c e) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )+\frac {1}{3} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {2 c^2 e \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \tan ^{-1}(c x)\right )^2}{3 b}-\frac {1}{3} b c^3 d \log (x)+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{12 e}-\frac {1}{6} (b c e) \text {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )\\ &=-\frac {2 c^2 e \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac {c^3 e \left (a+b \tan ^{-1}(c x)\right )^2}{3 b}-\frac {1}{3} b c^3 d \log (x)+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{12 e}+\frac {1}{6} b c^3 e \text {Li}_2\left (-c^2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 181, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (-\frac {8 c^2 e (a+b \text {ArcTan}(c x))}{x}-\frac {4 c^3 e (a+b \text {ArcTan}(c x))^2}{b}+6 b c^3 e \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )-\frac {2 b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2}-\frac {4 (a+b \text {ArcTan}(c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^3}+\frac {b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{e}-2 b c^3 \left (\log \left (-c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+e \text {PolyLog}\left (2,1+c^2 x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 20.84, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{4}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [A]
time = 47.91, size = 428, normalized size = 2.26 \begin {gather*} - \frac {2 a c^{2} e \operatorname {atan}{\left (\frac {x}{\sqrt {\frac {1}{c^{2}}}} \right )}}{3 \sqrt {\frac {1}{c^{2}}}} - \frac {2 a c^{2} e}{3 x} - \frac {a d}{3 x^{3}} - \frac {a e \log {\left (c^{2} x^{2} + 1 \right )}}{3 x^{3}} - 2 b c^{7} e \left (\begin {cases} \frac {x^{2}}{12 c^{2}} - \frac {\log {\left (c^{2} x^{2} + 1 \right )}}{12 c^{4}} & \text {for}\: c = 0 \\\frac {\log {\left (c^{2} x^{2} + 1 \right )}^{2}}{24 c^{4}} & \text {otherwise} \end {cases}\right ) + \frac {b c^{5} d \left (\begin {cases} x^{2} & \text {for}\: c^{2} = 0 \\\frac {\log {\left (c^{2} x^{2} + 1 \right )}}{c^{2}} & \text {otherwise} \end {cases}\right )}{6} + \frac {b c^{5} e \left (\begin {cases} x^{2} & \text {for}\: c^{2} = 0 \\\frac {\log {\left (c^{2} x^{2} + 1 \right )}}{c^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c^{2} x^{2} + 1 \right )}}{6} - \frac {b c^{3} d \log {\left (x^{2} \right )}}{6} + \frac {b c^{3} e \log {\left (x \right )}}{3} - \frac {b c^{3} e \log {\left (c^{2} x^{2} + 1 \right )}}{6} - \frac {b c^{3} e \log {\left (6 c^{2} \sqrt {\frac {1}{c^{2}}} + \frac {6 \sqrt {\frac {1}{c^{2}}}}{x^{2}} \right )}}{3} + \frac {b c^{3} e \operatorname {atan}^{2}{\left (\frac {x}{\sqrt {\frac {1}{c^{2}}}} \right )}}{3} + \frac {b c^{3} e \operatorname {Li}_{2}\left (c^{2} x^{2} e^{i \pi }\right )}{6} - \frac {2 b c^{2} e \operatorname {atan}{\left (c x \right )} \operatorname {atan}{\left (\frac {x}{\sqrt {\frac {1}{c^{2}}}} \right )}}{3 \sqrt {\frac {1}{c^{2}}}} - \frac {2 b c^{2} e \operatorname {atan}{\left (c x \right )}}{3 x} - \frac {b c d}{6 x^{2}} - \frac {b c e \log {\left (c^{2} x^{2} + 1 \right )}}{6 x^{2}} - \frac {b d \operatorname {atan}{\left (c x \right )}}{3 x^{3}} - \frac {b e \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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