Optimal. Leaf size=574 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^3}+\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a \log (x)}{d^3}+\frac {b c \sqrt {e} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {b c \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}+\frac {i b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^3}+\frac {i b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^3}+\frac {i b \text {Li}_2(-i c x)}{2 d^3}-\frac {i b \text {Li}_2(i c x)}{2 d^3}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^3} \]
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Rubi [A] time = 0.63, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4980, 4848, 2391, 4974, 414, 522, 203, 205, 391, 4856, 2402, 2315, 2447} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{4 d^3}+\frac {i b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{4 d^3}+\frac {i b \text {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \text {PolyLog}(2,i c x)}{2 d^3}-\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^3}+\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a \log (x)}{d^3}+\frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac {b c \sqrt {e} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {b c \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 391
Rule 414
Rule 522
Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 4848
Rule 4856
Rule 4974
Rule 4980
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )^3}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}-\frac {e 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} \, dx}{d^3}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^3}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^2}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{d}\\ &=\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a \log (x)}{d^3}+\frac {(i b) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^3}-\frac {(i b) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^3}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 d}-\frac {e \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 e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a \log (x)}{d^3}+\frac {i b \text {Li}_2(-i c x)}{2 d^3}-\frac {i b \text {Li}_2(i c x)}{2 d^3}-\frac {(b c) \int \frac {2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d^2 \left (c^2 d-e\right )}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}+\frac {\sqrt {e} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^3}-\frac {\sqrt {e} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^3}+\frac {(b c e) \int \frac {1}{d+e x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}\\ &=\frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {b c \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {a \log (x)}{d^3}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {\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^3}-\frac {\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^3}+\frac {i b \text {Li}_2(-i c x)}{2 d^3}-\frac {i b \text {Li}_2(i c x)}{2 d^3}-2 \frac {(b c) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^3}+\frac {(b c) \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^3}+\frac {(b c) \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^3}-\frac {\left (b c^5\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 d \left (c^2 d-e\right )^2}+\frac {\left (b c \left (3 c^2 d-e\right ) e\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 \left (c^2 d-e\right )^2}\\ &=\frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {b c \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c \left (3 c^2 d-e\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {a \log (x)}{d^3}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {\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^3}-\frac {\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^3}+\frac {i b \text {Li}_2(-i c x)}{2 d^3}-\frac {i b \text {Li}_2(i c x)}{2 d^3}+\frac {i b \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}+\frac {i b \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 \frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 d^3}\\ &=\frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^4 \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}-\frac {b c^2 \tan ^{-1}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac {a+b \tan ^{-1}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \tan ^{-1}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {b c \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c \left (3 c^2 d-e\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {a \log (x)}{d^3}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {\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^3}-\frac {\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^3}+\frac {i b \text {Li}_2(-i c x)}{2 d^3}-\frac {i b \text {Li}_2(i c x)}{2 d^3}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b \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}+\frac {i b \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}\\ \end {align*}
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Mathematica [A] time = 14.42, size = 645, normalized size = 1.12 \[ \frac {2 a \left (\frac {d \left (3 d+2 e x^2\right )}{\left (d+e x^2\right )^2}-2 \log \left (d+e x^2\right )+4 \log (x)\right )+b \left (\frac {c d e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {2 c^2 d \left (2 e-3 c^2 d\right ) \tan ^{-1}(c x)}{\left (e-c^2 d\right )^2}+\frac {c \sqrt {d} \sqrt {e} \left (7 c^2 d-5 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (e-c^2 d\right )^2}+2 i \text {Li}_2\left (\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )-2 i \text {Li}_2\left (\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {d} c+\sqrt {e}}\right )-2 i \text {Li}_2\left (\frac {c \left (i \sqrt {e} x+\sqrt {d}\right )}{c \sqrt {d}-\sqrt {e}}\right )+2 i \text {Li}_2\left (\frac {c \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {d} c+\sqrt {e}}\right )+\frac {2 d \tan ^{-1}(c x) \left (3 d+2 e x^2\right )}{\left (d+e x^2\right )^2}-2 i \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )+2 i \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )+2 i \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )-2 i \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )-4 \tan ^{-1}(c x) \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )-4 \tan ^{-1}(c x) \log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )-4 i (-\text {Li}_2(-i c x)+\text {Li}_2(i c x)+\log (x) (\log (1-i c x)-\log (1+i c x)))+8 \log (x) \tan ^{-1}(c x)\right )}{8 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} x}, x\right ) \]
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 = 0.36, size = 1041, normalized size = 1.81 \[ \text {result too large to display} \]
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 \[ \frac {1}{4} \, a {\left (\frac {2 \, e x^{2} + 3 \, d}{d^{2} e^{2} x^{4} + 2 \, d^{3} e x^{2} + d^{4}} - \frac {2 \, \log \left (e x^{2} + d\right )}{d^{3}} + \frac {4 \, \log \relax (x)}{d^{3}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} x\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]
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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