Integrand size = 19, antiderivative size = 83 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=-\frac {b c d}{6 x^2}-\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-\frac {1}{3} b c \left (c^2 d-3 e\right ) \log (x)+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (1+c^2 x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5096, 12, 457, 78} \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=-\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )-\frac {1}{3} b c \log (x) \left (c^2 d-3 e\right )-\frac {b c d}{6 x^2} \]
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 5096
Rubi steps \begin{align*} \text {integral}& = -\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-(b c) \int \frac {-d-3 e x^2}{3 x^3 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-\frac {1}{3} (b c) \int \frac {-d-3 e x^2}{x^3 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {-d-3 e x}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-\frac {1}{6} (b c) \text {Subst}\left (\int \left (-\frac {d}{x^2}+\frac {c^2 d-3 e}{x}+\frac {-c^4 d+3 c^2 e}{1+c^2 x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b c d}{6 x^2}-\frac {d (a+b \arctan (c x))}{3 x^3}-\frac {e (a+b \arctan (c x))}{x}-\frac {1}{3} b c \left (c^2 d-3 e\right ) \log (x)+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (1+c^2 x^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=-\frac {a d}{3 x^3}-\frac {a e}{x}-\frac {b d \arctan (c x)}{3 x^3}-\frac {b e \arctan (c x)}{x}+b c e \log (x)-\frac {1}{2} b c e \log \left (1+c^2 x^2\right )+\frac {1}{6} b c d \left (-\frac {1}{x^2}-2 c^2 \log (x)+c^2 \log \left (1+c^2 x^2\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17
method | result | size |
parts | \(a \left (-\frac {e}{x}-\frac {d}{3 x^{3}}\right )+b \,c^{3} \left (-\frac {\arctan \left (c x \right ) e}{c^{3} x}-\frac {\arctan \left (c x \right ) d}{3 c^{3} x^{3}}-\frac {\left (c^{2} d -3 e \right ) \ln \left (c x \right )+\frac {d}{2 x^{2}}+\frac {\left (-c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{3 c^{2}}\right )\) | \(97\) |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {e}{c x}-\frac {d}{3 c \,x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) e}{c x}-\frac {\arctan \left (c x \right ) d}{3 c \,x^{3}}-\frac {\left (-c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\left (c^{2} d -3 e \right ) \ln \left (c x \right )}{3}-\frac {d}{6 x^{2}}\right )}{c^{2}}\right )\) | \(105\) |
default | \(c^{3} \left (\frac {a \left (-\frac {e}{c x}-\frac {d}{3 c \,x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) e}{c x}-\frac {\arctan \left (c x \right ) d}{3 c \,x^{3}}-\frac {\left (-c^{2} d +3 e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\left (c^{2} d -3 e \right ) \ln \left (c x \right )}{3}-\frac {d}{6 x^{2}}\right )}{c^{2}}\right )\) | \(105\) |
parallelrisch | \(-\frac {2 \ln \left (x \right ) b \,c^{3} d \,x^{3}-\ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,x^{3}-b \,c^{3} d \,x^{3}-6 \ln \left (x \right ) b c e \,x^{3}+3 \ln \left (c^{2} x^{2}+1\right ) b c e \,x^{3}+6 \arctan \left (c x \right ) b e \,x^{2}+6 a e \,x^{2}+b c d x +2 \arctan \left (c x \right ) b d +2 a d}{6 x^{3}}\) | \(112\) |
risch | \(\frac {i b \left (3 e \,x^{2}+d \right ) \ln \left (i c x +1\right )}{6 x^{3}}-\frac {2 \ln \left (x \right ) b \,c^{3} d \,x^{3}-\ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,x^{3}-6 \ln \left (x \right ) b c e \,x^{3}+3 \ln \left (c^{2} x^{2}+1\right ) b c e \,x^{3}+3 i b e \ln \left (-i c x +1\right ) x^{2}+i b d \ln \left (-i c x +1\right )+6 a e \,x^{2}+b c d x +2 a d}{6 x^{3}}\) | \(136\) |
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Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=\frac {{\left (b c^{3} d - 3 \, b c e\right )} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{3} d - 3 \, b c e\right )} x^{3} \log \left (x\right ) - b c d x - 6 \, a e x^{2} - 2 \, a d - 2 \, {\left (3 \, b e x^{2} + b d\right )} \arctan \left (c x\right )}{6 \, x^{3}} \]
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Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=\begin {cases} - \frac {a d}{3 x^{3}} - \frac {a e}{x} - \frac {b c^{3} d \log {\left (x \right )}}{3} + \frac {b c^{3} d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6} - \frac {b c d}{6 x^{2}} + b c e \log {\left (x \right )} - \frac {b c e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d \operatorname {atan}{\left (c x \right )}}{3 x^{3}} - \frac {b e \operatorname {atan}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\a \left (- \frac {d}{3 x^{3}} - \frac {e}{x}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
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Time = 0.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d+e x^2\right ) (a+b \arctan (c x))}{x^4} \, dx=b\,c\,e\,\ln \left (x\right )-\frac {a\,e}{x}-\frac {b\,c\,e\,\ln \left (c^2\,x^2+1\right )}{2}-\frac {b\,c\,d}{6\,x^2}-\frac {a\,d}{3\,x^3}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{3\,x^3}-\frac {b\,e\,\mathrm {atan}\left (c\,x\right )}{x}+\frac {b\,c^3\,d\,\ln \left (c^2\,x^2+1\right )}{6}-\frac {b\,c^3\,d\,\ln \left (x\right )}{3} \]
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