Integrand size = 19, antiderivative size = 131 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}} \]
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Time = 0.07 (sec) , antiderivative size = 131, 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 = {5094, 425, 536, 209, 211} \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \sqrt {e} \left (c^2 d-e\right )^2}+\frac {b c^4 \arctan (c x)}{4 e \left (c^2 d-e\right )^2}-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )} \]
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Rule 209
Rule 211
Rule 425
Rule 536
Rule 5094
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d \left (c^2 d-e\right ) e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b c \left (3 c^2 d-e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d \left (c^2 d-e\right )^2}+\frac {\left (b c^5\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 \left (c^2 d-e\right )^2 e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \left (d+e x^2\right )}{d \left (c^2 d-e\right )}}{\left (d+e x^2\right )^2}+\frac {2 b \left (\frac {c^4}{\left (-c^2 d+e\right )^2}-\frac {1}{\left (d+e x^2\right )^2}\right ) \arctan (c x)}{e}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e} \left (-c^2 d+e\right )^2}\right ) \]
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Time = 0.70 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.20
method | result | size |
parts | \(-\frac {a}{4 e \left (e \,x^{2}+d \right )^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {c^{6} \left (-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}\right )}{4 e}\right )}{c^{2}}\) | \(157\) |
derivativedivides | \(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e}\right )}{c^{2}}\) | \(165\) |
default | \(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e}\right )}{c^{2}}\) | \(165\) |
risch | \(\frac {i b \ln \left (i c x +1\right )}{8 e \left (e \,x^{2}+d \right )^{2}}+\frac {c^{5} b x}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\frac {c^{3} b e x}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right ) d}+\frac {i c^{6} b \ln \left (-i c x +1\right ) d}{4 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}+\frac {i c b \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right ) e}{16 \left (c^{2} d -e \right )^{2} d \sqrt {e d}}-\frac {3 i c^{3} b \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {e d}}-\frac {i c^{4} b \ln \left (-i c x +1\right ) e}{8 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}+\frac {i c^{8} b \ln \left (-i c x +1\right ) d \,x^{2}}{4 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}-\frac {i b \,c^{4} d}{16 e \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {i b \,c^{2}}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {i c^{8} b \ln \left (-i c x +1\right ) e \,x^{4}}{8 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}-\frac {i c^{4} b \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{16 \left (c^{2} d -e \right )^{2} e}-\frac {c^{4} a}{4 e \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2}}+\frac {i c^{4} b}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}+\frac {b \,c^{4} \arctan \left (c x \right )}{8 \left (c^{2} d -e \right )^{2} e}-\frac {b \,c^{3} x}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {e b c x}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right ) d}-\frac {i b \,c^{4} \ln \left (c^{2} x^{2}+1\right )}{16 e \left (c^{2} d -e \right )^{2}}-\frac {i c^{6} b d}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right ) e}+\frac {i b \,c^{4} \ln \left (e \,x^{2}+d \right )}{16 e \left (c^{2} d -e \right )^{2}}-\frac {3 b \,c^{3} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {e d}}+\frac {e b c \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} d \sqrt {e d}}\) | \(802\) |
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (115) = 230\).
Time = 0.35 (sec) , antiderivative size = 637, normalized size of antiderivative = 4.86 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} - {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 4 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 2 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Time = 2.88 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.53 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b\,c\,x}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e\,x^2+d\right )}^2}-\frac {a}{4\,e\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c^4\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e-c^2\,d\right )}^2}+\frac {b\,c\,e\,x^3}{8\,d\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{8\,d^3\,{\left (e-c^2\,d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,3{}\mathrm {i}}{8\,d^2\,e\,{\left (e-c^2\,d\right )}^2} \]
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