Integrand size = 21, antiderivative size = 457 \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\frac {c^2 (a+b \arctan (c x))^2}{2 \left (c^2 d-e\right ) e}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {b c (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {b c (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}-\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}} \]
[Out]
Time = 0.76 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5098, 4974, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964} \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\frac {c^2 (a+b \arctan (c x))^2}{2 e \left (c^2 d-e\right )}-\frac {b c (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}+\frac {b c (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}+\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )} \]
[In]
[Out]
Rule 2352
Rule 2449
Rule 2497
Rule 4964
Rule 4966
Rule 4974
Rule 5004
Rule 5040
Rule 5098
Rule 5104
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+b \arctan (c x))^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt {e}}-\frac {\int \frac {(a+b \arctan (c x))^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt {e}} \\ & = -\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(b c) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 d \left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{\sqrt {-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d e}+\frac {(b c) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d e} \\ & = -\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(b c) \int \frac {a+b \arctan (c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} \left (c^2 d-e\right )}+\frac {(b c) \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} \left (c^2 d-e\right )}+\frac {\left (b c^3\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 \sqrt {-d} \left (c^2 d-e\right ) e}+\frac {\left (b c^3\right ) \int \frac {\left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right ) e} \\ & = -\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {\left (b c^3\right ) \int \left (\frac {\sqrt {-d} (a+b \arctan (c x))}{1+c^2 x^2}+\frac {\sqrt {e} x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 \sqrt {-d} \left (c^2 d-e\right ) e}+\frac {\left (b c^3\right ) \int \left (\frac {d (a+b \arctan (c x))}{1+c^2 x^2}+\frac {\sqrt {-d} \sqrt {e} x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 d \left (c^2 d-e\right ) e}+\frac {\left (b^2 c^2\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}-\frac {\left (b^2 c^2\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}} \\ & = -\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {i b^2 c \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}-\frac {i b^2 c \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+2 \frac {\left (b c^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e} \\ & = \frac {c^2 (a+b \arctan (c x))^2}{2 \left (c^2 d-e\right ) e}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {b c (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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+\frac {i b^2 c \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}-\frac {i b^2 c \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}} \\ \end{align*}
Time = 6.93 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.83 \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\frac {1}{4} \left (-\frac {2 a^2}{e \left (d+e x^2\right )}+\frac {4 a b \left (-\frac {\left (1+c^2 x^2\right ) \arctan (c x)}{d+e x^2}+\frac {c \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}\right )}{-c^2 d+e}+\frac {b^2 c^2 \left (\frac {4 \arctan (c x)^2}{c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}+\frac {-4 \arctan (c x) \text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )-2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c d+\sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c d+\sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )-i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c d-\sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c d+\sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c d-\sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c d+\sqrt {-c^2 d e} x\right )}\right )\right )}{\sqrt {-c^2 d e}}\right )}{c^2 d-e}\right ) \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (377 ) = 754\).
Time = 2.58 (sec) , antiderivative size = 1175, normalized size of antiderivative = 2.57
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1175\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1209\) |
default | \(\text {Expression too large to display}\) | \(1209\) |
[In]
[Out]
\[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]
[In]
[Out]