Optimal. Leaf size=1039 \[ \frac {i c (a+b \text {ArcTan}(c x))^2}{2 d \left (c^2 d-e\right )}-\frac {(a+b \text {ArcTan}(c x))^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {(a+b \text {ArcTan}(c x))^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c (a+b \text {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 d \left (c^2 d-e\right )}-\frac {(a+b \text {ArcTan}(c x))^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c (a+b \text {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 d \left (c^2 d-e\right )}+\frac {(a+b \text {ArcTan}(c x))^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b^2 c \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}+\frac {i b^2 c \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {i b^2 c \text {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 d \left (c^2 d-e\right )}+\frac {i b (a+b \text {ArcTan}(c x)) \text {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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {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 d \left (c^2 d-e\right )}-\frac {i b (a+b \text {ArcTan}(c x)) \text {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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.99, antiderivative size = 1039, normalized size of antiderivative = 1.00, number of steps
used = 32, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5034, 4974,
4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964, 4968} \begin {gather*} \frac {i c \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b^2}{2 d \left (c^2 d-e\right )}+\frac {i c \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) b^2}{2 d \left (c^2 d-e\right )}-\frac {i c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 d \left (c^2 d-e\right )}-\frac {i c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 d \left (c^2 d-e\right )}-\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{3/2} \sqrt {e}}+\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{3/2} \sqrt {e}}-\frac {c (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right ) b}{d \left (c^2 d-e\right )}+\frac {c (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{i c x+1}\right ) b}{d \left (c^2 d-e\right )}+\frac {c (a+b \text {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 ) b}{2 d \left (c^2 d-e\right )}+\frac {c (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 d \left (c^2 d-e\right )}+\frac {i (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 (-d)^{3/2} \sqrt {e}}-\frac {i (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 (-d)^{3/2} \sqrt {e}}+\frac {i c (a+b \text {ArcTan}(c x))^2}{2 d \left (c^2 d-e\right )}-\frac {(a+b \text {ArcTan}(c x))^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {(a+b \text {ArcTan}(c x))^2}{4 d \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {(a+b \text {ArcTan}(c x))^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{3/2} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2352
Rule 2449
Rule 2497
Rule 4964
Rule 4966
Rule 4968
Rule 4974
Rule 5004
Rule 5034
Rule 5040
Rule 5104
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {e \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {(b c) \int \left (\frac {\sqrt {e} \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {c^2 \left (-\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d}+\frac {(b c) \int \left (-\frac {\sqrt {e} \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d}-\frac {e \int \left (-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2}}-\frac {\left (b c^3\right ) \int \frac {\left (-\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right ) \sqrt {e}}-\frac {\left (b c^3\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right ) \sqrt {e}}+\frac {\left (b c \sqrt {e}\right ) \int \frac {a+b \tan ^{-1}(c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 d \left (c^2 d-e\right )}+\frac {\left (b c \sqrt {e}\right ) \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+2 \frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}-\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 d \left (c^2 d-e\right )}-\frac {\left (b c^3\right ) \int \left (-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 d \left (c^2 d-e\right ) \sqrt {e}}-\frac {\left (b c^3\right ) \int \left (\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 d \left (c^2 d-e\right ) \sqrt {e}}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+2 \frac {\left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-2 \frac {\left (b c^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}-2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac {\left (b c^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}-2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}+\frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}-2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {\left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}\right )\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \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 d \left (c^2 d-e\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \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 )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d \left (c^2 d-e\right )}-2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (c^2 d-e\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 d \left (c^2 d-e\right )}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{4 d \left (c^2 d-e\right )}\right )-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {i b^2 c \text {Li}_2\left (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 d \left (c^2 d-e\right )}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 (-d)^{3/2} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [F]
time = 17.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcTan}(c x))^2}{\left (d+e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 179.98, size = 6570, normalized size = 6.32
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(6570\) |
default | \(\text {Expression too large to display}\) | \(6570\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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