Optimal. Leaf size=489 \[ -\frac {2 e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{d^3}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{d^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {2 a e \log (x)}{d^3}-\frac {b c e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c^2 e \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2}-\frac {i b e \text {Li}_2(-i c x)}{d^3}+\frac {i b e \text {Li}_2(i c x)}{d^3}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {b c}{2 d^2 x} \]
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Rubi [A] time = 0.51, antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4980, 4852, 325, 203, 4848, 2391, 4974, 391, 205, 4856, 2402, 2315, 2447} \[ -\frac {i b e \text {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \text {PolyLog}(2,i c x)}{d^3}+\frac {i b e \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^3}-\frac {i b e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {2 e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{d^3}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{d^3}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {2 a e \log (x)}{d^3}-\frac {b c e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c^2 e \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2}-\frac {b c}{2 d^2 x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 325
Rule 391
Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 4848
Rule 4852
Rule 4856
Rule 4974
Rule 4980
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d^2 x^3}-\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^3 x}+\frac {e^2 x \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}+\frac {2 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^3}+\frac {e^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^2}\\ &=-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {2 a e \log (x)}{d^3}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d^2}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{d^3}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2}+\frac {\left (2 e^2\right ) \int \left (-\frac {a+b \tan ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \tan ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^3}\\ &=-\frac {b c}{2 d^2 x}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {2 a e \log (x)}{d^3}-\frac {i b e \text {Li}_2(-i c x)}{d^3}+\frac {i b e \text {Li}_2(i c x)}{d^3}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2}+\frac {\left (b c^3 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}-\frac {e^{3/2} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{d^3}+\frac {e^{3/2} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{d^3}-\frac {\left (b c e^2\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}\\ &=-\frac {b c}{2 d^2 x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2}+\frac {b c^2 e \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e \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 )}{d^3}+\frac {e \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 )}{d^3}-\frac {i b e \text {Li}_2(-i c x)}{d^3}+\frac {i b e \text {Li}_2(i c x)}{d^3}+2 \frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {(b c e) \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}{d^3}-\frac {(b c e) \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}{d^3}\\ &=-\frac {b c}{2 d^2 x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2}+\frac {b c^2 e \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e \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 )}{d^3}+\frac {e \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 )}{d^3}-\frac {i b e \text {Li}_2(-i c x)}{d^3}+\frac {i b e \text {Li}_2(i c x)}{d^3}-\frac {i b e \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 )}{2 d^3}-\frac {i b e \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 )}{2 d^3}+2 \frac {(i b e) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d^3}\\ &=-\frac {b c}{2 d^2 x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2}+\frac {b c^2 e \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e \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 )}{d^3}+\frac {e \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 )}{d^3}-\frac {i b e \text {Li}_2(-i c x)}{d^3}+\frac {i b e \text {Li}_2(i c x)}{d^3}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \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 )}{2 d^3}-\frac {i b e \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 )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 13.70, size = 643, normalized size = 1.31 \[ -\frac {a \left (d \left (\frac {e}{d+e x^2}+\frac {1}{x^2}\right )-2 e \log \left (d+e x^2\right )+4 e \log (x)\right )+b \left (\frac {c \sqrt {d} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{c^2 d-e}+\frac {c^2 d \left (c^2 d-2 e\right ) \tan ^{-1}(c x)}{c^2 d-e}-e \left (-i \text {Li}_2\left (\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )+i \text {Li}_2\left (\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {d} c+\sqrt {e}}\right )+i \text {Li}_2\left (\frac {c \left (i \sqrt {e} x+\sqrt {d}\right )}{c \sqrt {d}-\sqrt {e}}\right )-i \text {Li}_2\left (\frac {c \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {d} c+\sqrt {e}}\right )-2 \tan ^{-1}(c x) \log \left (d+e x^2\right )+i \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )-i \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )+i \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )+2 \tan ^{-1}(c x) \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )+2 \tan ^{-1}(c x) \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )\right )+d \tan ^{-1}(c x) \left (\frac {e}{d+e x^2}+\frac {1}{x^2}\right )-2 e \tan ^{-1}(c x) \log \left (d+e x^2\right )+\frac {c d}{x}-2 i e (-\text {Li}_2(-i c x)+\text {Li}_2(i c x)+\log (x) (\log (1-i c x)-\log (1+i c x)))+4 e \log (x) \tan ^{-1}(c x)\right )}{2 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.37, size = 925, normalized size = 1.89 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {2 \, e x^{2} + d}{d^{2} e x^{4} + d^{3} x^{2}} - \frac {2 \, e \log \left (e x^{2} + d\right )}{d^{3}} + \frac {4 \, e \log \relax (x)}{d^{3}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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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