Optimal. Leaf size=457 \[ \frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d-e\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )} \]
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Rubi [A] time = 1.09, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854} \[ \frac {i b^2 c \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {i b^2 c \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{4 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}+\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d-e\right )}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 \sqrt {-d} \sqrt {e} \left (c^2 d-e\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 4854
Rule 4856
Rule 4864
Rule 4884
Rule 4920
Rule 4978
Rule 4984
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt {e}}-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{3/2} \sqrt {e}}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(b c) \int \left (\frac {\sqrt {-d} 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 d \left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\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 \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 (d+\sqrt {-d} \sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(b c) \int \frac {a+b \tan ^{-1}(c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} \left (c^2 d-e\right )}+\frac {(b c) \int \frac {a+b \tan ^{-1}(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 ) \left (a+b \tan ^{-1}(c x)\right )}{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 ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right ) e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-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 \sqrt {-d} \left (c^2 d-e\right ) e}+\frac {\left (b c^3\right ) \int \left (\frac {d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {-d} \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 ) 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 {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}+2 \frac {\left (b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d e \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \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 \sqrt {-d} \left (c^2 d-e\right ) \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 8.88, size = 885, normalized size = 1.94 \[ -\frac {a^2}{2 e \left (e x^2+d\right )}+2 b c^2 \left (\frac {c \tan ^{-1}(c x)-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}}{2 e \left (c^3 d-c e\right )}-\frac {\tan ^{-1}(c x)}{2 e \left (e x^2 c^2+d c^2\right )}\right ) a+\frac {b^2 c^2 \left (\frac {4 \tan ^{-1}(c x)^2}{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}+\frac {4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )-2 \cos ^{-1}\left (-\frac {d c^2+e}{c^2 d-e}\right ) \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )+\left (\cos ^{-1}\left (-\frac {d c^2+e}{c^2 d-e}\right )+2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (\sqrt {-c^2 d e}-i e\right ) (c x-i)}{\left (c^2 d-e\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}\right )+\left (\cos ^{-1}\left (-\frac {d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (c x+i)}{\left (c^2 d-e\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}\right )-\left (\cos ^{-1}\left (-\frac {d c^2+e}{c^2 d-e}\right )-2 i \left (\tanh ^{-1}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\tanh ^{-1}\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 \tan ^{-1}(c x)}}{\sqrt {c^2 d-e} \sqrt {d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-\left (\cos ^{-1}\left (-\frac {d c^2+e}{c^2 d-e}\right )+2 i \left (\tanh ^{-1}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\tanh ^{-1}\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 \tan ^{-1}(c x)}}{\sqrt {c^2 d-e} \sqrt {d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-i \left (\text {Li}_2\left (\frac {\left (d c^2+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}\right )-\text {Li}_2\left (\frac {\left (d c^2+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}\right )\right )}{\sqrt {-c^2 d e}}\right )}{4 \left (c^2 d-e\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x \arctan \left (c x\right )^{2} + 2 \, a b x \arctan \left (c x\right ) + a^{2} x}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.05, size = 1185, normalized size = 2.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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