Optimal. Leaf size=574 \[ \frac {b c e x}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^4 \text {ArcTan}(c x)}{4 d \left (c^2 d-e\right )^2}-\frac {b c^2 \text {ArcTan}(c x)}{2 d^2 \left (c^2 d-e\right )}+\frac {a+b \text {ArcTan}(c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \text {ArcTan}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {b c \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c \left (3 c^2 d-e\right ) \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {a \log (x)}{d^3}+\frac {(a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {(a+b \text {ArcTan}(c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {(a+b \text {ArcTan}(c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}+\frac {i b \text {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \text {PolyLog}(2,i c x)}{2 d^3}-\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^3}+\frac {i b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^3} \]
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Rubi [A]
time = 0.45, 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 = {5100, 4940,
2438, 5094, 425, 536, 209, 211, 400, 4966, 2449, 2352, 2497} \begin {gather*} -\frac {(a+b \text {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 d^3}-\frac {(a+b \text {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 d^3}+\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))}{d^3}+\frac {a+b \text {ArcTan}(c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a+b \text {ArcTan}(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 ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \left (c^2 d-e\right )^2}+\frac {b c \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {b c^2 \text {ArcTan}(c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {b c^4 \text {ArcTan}(c x)}{4 d \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 {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 400
Rule 425
Rule 536
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 4940
Rule 4966
Rule 5094
Rule 5100
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) \text {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 = 9.47, size = 645, normalized size = 1.12 \begin {gather*} \frac {2 a \left (\frac {d \left (3 d+2 e x^2\right )}{\left (d+e x^2\right )^2}+4 \log (x)-2 \log \left (d+e x^2\right )\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 (-3 c^2 d+2 e\right ) \text {ArcTan}(c x)}{\left (-c^2 d+e\right )^2}+\frac {2 d \left (3 d+2 e x^2\right ) \text {ArcTan}(c x)}{\left (d+e x^2\right )^2}+\frac {c \sqrt {d} \left (7 c^2 d-5 e\right ) \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (-c^2 d+e\right )^2}+8 \text {ArcTan}(c x) \log (x)-4 \text {ArcTan}(c x) \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )-4 \text {ArcTan}(c x) \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )-2 i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )+2 i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )+2 i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )-2 i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )-4 i (\log (x) (\log (1-i c x)-\log (1+i c x))-\text {PolyLog}(2,-i c x)+\text {PolyLog}(2,i c x))+2 i \text {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )-2 i \text {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )-2 i \text {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )+2 i \text {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )}{8 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.21, size = 1041, normalized size = 1.81
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1041\) |
default | \(\text {Expression too large to display}\) | \(1041\) |
risch | \(\text {Expression too large to display}\) | \(2236\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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