Integrand size = 18, antiderivative size = 501 \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{2 \sqrt {-c^2} e} \]
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Time = 0.64 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4976, 281, 209, 2463, 266, 2441, 2440, 2438} \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {\log (d+e x) \left (a+b \arctan \left (c x^2\right )\right )}{e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [4]{-c^2} x+1\right )}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt {-\sqrt {-c^2}} x+1\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{2 \sqrt {-c^2} e} \]
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Rule 209
Rule 266
Rule 281
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 4976
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {(2 b c) \int \frac {x \log (d+e x)}{1+c^2 x^4} \, dx}{e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {(2 b c) \int \left (-\frac {c^2 x \log (d+e x)}{2 \sqrt {-c^2} \left (\sqrt {-c^2}-c^2 x^2\right )}-\frac {c^2 x \log (d+e x)}{2 \sqrt {-c^2} \left (\sqrt {-c^2}+c^2 x^2\right )}\right ) \, dx}{e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {\left (b c \sqrt {-c^2}\right ) \int \frac {x \log (d+e x)}{\sqrt {-c^2}-c^2 x^2} \, dx}{e}-\frac {\left (b c \sqrt {-c^2}\right ) \int \frac {x \log (d+e x)}{\sqrt {-c^2}+c^2 x^2} \, dx}{e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {\left (b c \sqrt {-c^2}\right ) \int \left (-\frac {\sqrt [4]{-c^2} \log (d+e x)}{2 c^2 \left (1-\sqrt [4]{-c^2} x\right )}+\frac {\sqrt [4]{-c^2} \log (d+e x)}{2 c^2 \left (1+\sqrt [4]{-c^2} x\right )}\right ) \, dx}{e}-\frac {\left (b c \sqrt {-c^2}\right ) \int \left (\frac {\sqrt {-\sqrt {-c^2}} \log (d+e x)}{2 c^2 \left (1-\sqrt {-\sqrt {-c^2}} x\right )}-\frac {\sqrt {-\sqrt {-c^2}} \log (d+e x)}{2 c^2 \left (1+\sqrt {-\sqrt {-c^2}} x\right )}\right ) \, dx}{e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {(b c) \int \frac {\log (d+e x)}{1-\sqrt [4]{-c^2} x} \, dx}{2 \sqrt [4]{-c^2} e}+\frac {(b c) \int \frac {\log (d+e x)}{1+\sqrt [4]{-c^2} x} \, dx}{2 \sqrt [4]{-c^2} e}-\frac {(b c) \int \frac {\log (d+e x)}{1-\sqrt {-\sqrt {-c^2}} x} \, dx}{2 \sqrt {-\sqrt {-c^2}} e}+\frac {(b c) \int \frac {\log (d+e x)}{1+\sqrt {-\sqrt {-c^2}} x} \, dx}{2 \sqrt {-\sqrt {-c^2}} e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {(b c) \int \frac {\log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right )}{d+e x} \, dx}{2 \sqrt {-c^2}}-\frac {(b c) \int \frac {\log \left (\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{-\sqrt [4]{-c^2} d+e}\right )}{d+e x} \, dx}{2 \sqrt {-c^2}}+\frac {(b c) \int \frac {\log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{d+e x} \, dx}{2 \sqrt {-c^2}}+\frac {(b c) \int \frac {\log \left (\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{-\sqrt {-\sqrt {-c^2}} d+e}\right )}{d+e x} \, dx}{2 \sqrt {-c^2}} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {(b c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{-c^2} x}{-\sqrt [4]{-c^2} d+e}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-c^2} e}-\frac {(b c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{-c^2} x}{\sqrt [4]{-c^2} d+e}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-c^2} e}+\frac {(b c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-\sqrt {-c^2}} x}{-\sqrt {-\sqrt {-c^2}} d+e}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-c^2} e}+\frac {(b c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-\sqrt {-c^2}} x}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-c^2} e} \\ & = \frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{2 \sqrt {-c^2} e} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.06 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \arctan \left (c x^2\right ) \log (d+e x)+i \left (\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt [4]{-1} e}\right )+\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt [4]{-1} e}\right )-\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-(-1)^{3/4} e}\right )-\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+(-1)^{3/4} e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt [4]{-1} e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt [4]{-1} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-(-1)^{3/4} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+(-1)^{3/4} e}\right )\right )\right )}{2 e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.28
method | result | size |
default | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{2}\right )}{e}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}-4 c^{2} d \,\textit {\_Z}^{3}+6 c^{2} d^{2} \textit {\_Z}^{2}-4 \textit {\_Z} \,c^{2} d^{3}+c^{2} d^{4}+e^{4}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} d +d^{2}}\right )}{2 c}\) | \(138\) |
parts | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{2}\right )}{e}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}-4 c^{2} d \,\textit {\_Z}^{3}+6 c^{2} d^{2} \textit {\_Z}^{2}-4 \textit {\_Z} \,c^{2} d^{3}+c^{2} d^{4}+e^{4}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} d +d^{2}}\right )}{2 c}\) | \(138\) |
risch | \(\frac {i b \ln \left (e x +d \right ) \ln \left (-i c \,x^{2}+1\right )}{2 e}-\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-i c}-c \left (e x +d \right )+c d}{e \sqrt {-i c}+c d}\right )}{2 e}-\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-i c}+c \left (e x +d \right )-c d}{e \sqrt {-i c}-c d}\right )}{2 e}-\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {-i c}-c \left (e x +d \right )+c d}{e \sqrt {-i c}+c d}\right )}{2 e}-\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {-i c}+c \left (e x +d \right )-c d}{e \sqrt {-i c}-c d}\right )}{2 e}+\frac {a \ln \left (e x +d \right )}{e}-\frac {i b \ln \left (e x +d \right ) \ln \left (i c \,x^{2}+1\right )}{2 e}+\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {i c}-c \left (e x +d \right )+c d}{e \sqrt {i c}+c d}\right )}{2 e}+\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {i c}+c \left (e x +d \right )-c d}{e \sqrt {i c}-c d}\right )}{2 e}+\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {i c}-c \left (e x +d \right )+c d}{e \sqrt {i c}+c d}\right )}{2 e}+\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {i c}+c \left (e x +d \right )-c d}{e \sqrt {i c}-c d}\right )}{2 e}\) | \(431\) |
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\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \]
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\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x^2\right )}{d+e\,x} \,d x \]
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