Integrand size = 21, antiderivative size = 532 \[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c d x}{8 \left (c^2 d-e\right ) e^2 \left (d+e x^2\right )}+\frac {b c^4 d^2 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e^3}-\frac {b c^2 d \arctan (c x)}{\left (c^2 d-e\right ) e^3}-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c^2 d-e\right ) e^{5/2}}-\frac {b c \sqrt {d} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \left (c^2 d-e\right )^2 e^{5/2}}-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3} \]
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Time = 0.45 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5100, 5094, 425, 536, 209, 211, 400, 4966, 2449, 2352, 2497} \[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 e^3}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^3}-\frac {b c \sqrt {d} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 e^{5/2} \left (c^2 d-e\right )^2}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c^2 d-e\right )}-\frac {b c^2 d \arctan (c x)}{e^3 \left (c^2 d-e\right )}+\frac {b c^4 d^2 \arctan (c x)}{4 e^3 \left (c^2 d-e\right )^2}-\frac {b c d x}{8 e^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3} \]
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Rule 209
Rule 211
Rule 400
Rule 425
Rule 536
Rule 2352
Rule 2449
Rule 2497
Rule 4966
Rule 5094
Rule 5100
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 x (a+b \arctan (c x))}{e^2 \left (d+e x^2\right )^3}-\frac {2 d x (a+b \arctan (c x))}{e^2 \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \arctan (c x))}{d+e x^2} \, dx}{e^2}-\frac {(2 d) \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac {d^2 \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx}{e^2} \\ & = -\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{e^3}+\frac {\left (b c d^2\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac {\int \left (-\frac {a+b \arctan (c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arctan (c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e^2} \\ & = -\frac {b c d x}{8 \left (c^2 d-e\right ) e^2 \left (d+e x^2\right )}-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {(b c d) \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 \left (c^2 d-e\right ) e^3}-\frac {\left (b c^3 d\right ) \int \frac {1}{1+c^2 x^2} \, dx}{\left (c^2 d-e\right ) e^3}-\frac {\int \frac {a+b \arctan (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{5/2}}+\frac {\int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{5/2}}+\frac {(b c d) \int \frac {1}{d+e x^2} \, dx}{\left (c^2 d-e\right ) e^2} \\ & = -\frac {b c d x}{8 \left (c^2 d-e\right ) e^2 \left (d+e x^2\right )}-\frac {b c^2 d \arctan (c x)}{\left (c^2 d-e\right ) e^3}-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c^2 d-e\right ) e^{5/2}}-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+2 \frac {(b c) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 e^3}-\frac {(b c) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^3}-\frac {(b c) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^3}+\frac {\left (b c^5 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 \left (c^2 d-e\right )^2 e^3}-\frac {\left (b c d \left (3 c^2 d-e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 \left (c^2 d-e\right )^2 e^2} \\ & = -\frac {b c d x}{8 \left (c^2 d-e\right ) e^2 \left (d+e x^2\right )}+\frac {b c^4 d^2 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e^3}-\frac {b c^2 d \arctan (c x)}{\left (c^2 d-e\right ) e^3}-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c^2 d-e\right ) e^{5/2}}-\frac {b c \sqrt {d} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \left (c^2 d-e\right )^2 e^{5/2}}-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}+2 \frac {(i b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 e^3} \\ & = -\frac {b c d x}{8 \left (c^2 d-e\right ) e^2 \left (d+e x^2\right )}+\frac {b c^4 d^2 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e^3}-\frac {b c^2 d \arctan (c x)}{\left (c^2 d-e\right ) e^3}-\frac {d^2 (a+b \arctan (c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \arctan (c x))}{e^3 \left (d+e x^2\right )}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c^2 d-e\right ) e^{5/2}}-\frac {b c \sqrt {d} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \left (c^2 d-e\right )^2 e^{5/2}}-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^3} \\ \end{align*}
Time = 9.84 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.11 \[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {a \left (\frac {d \left (3 d+4 e x^2\right )}{\left (d+e x^2\right )^2}+2 \log \left (d+e x^2\right )\right )+b \left (-\frac {c d e x}{2 \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {c^2 d \left (-3 c^2 d+4 e\right ) \arctan (c x)}{\left (-c^2 d+e\right )^2}+\frac {d \left (3 d+4 e x^2\right ) \arctan (c x)}{\left (d+e x^2\right )^2}+\frac {c \sqrt {d} \left (5 c^2 d-7 e\right ) \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \left (-c^2 d+e\right )^2}+2 \arctan (c x) \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+2 \arctan (c x) \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )-i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )+i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )}{4 e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.66 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.54
method | result | size |
parts | \(\text {Expression too large to display}\) | \(817\) |
derivativedivides | \(\text {Expression too large to display}\) | \(843\) |
default | \(\text {Expression too large to display}\) | \(843\) |
risch | \(\text {Expression too large to display}\) | \(1666\) |
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\[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^5 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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