Optimal. Leaf size=183 \[ -\frac {3 b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {3 i b \sqrt {e} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {3 i b \sqrt {e} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2385, 2380,
2341, 211, 2361, 12, 4940, 2438} \begin {gather*} \frac {3 i b \sqrt {e} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {3 i b \sqrt {e} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {\sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )}{2 d^{5/2}}-\frac {3 a+3 b \log \left (c x^n\right )-b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 b n}{2 d^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2341
Rule 2361
Rule 2380
Rule 2385
Rule 2438
Rule 4940
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^2} \, dx &=\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {\int \frac {-3 a+b n-3 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx}{2 d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {\int \left (\frac {-3 a+b n-3 b \log \left (c x^n\right )}{d x^2}-\frac {e \left (-3 a+b n-3 b \log \left (c x^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx}{2 d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {\int \frac {-3 a+b n-3 b \log \left (c x^n\right )}{x^2} \, dx}{2 d^2}+\frac {e \int \frac {-3 a+b n-3 b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d^2}\\ &=-\frac {3 b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {(3 b e n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 d^2}\\ &=-\frac {3 b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {\left (3 b \sqrt {e} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 d^{5/2}}\\ &=-\frac {3 b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {\left (3 i b \sqrt {e} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{5/2}}-\frac {\left (3 i b \sqrt {e} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{5/2}}\\ &=-\frac {3 b n}{2 d^2 x}+\frac {a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac {3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac {3 i b \sqrt {e} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}}-\frac {3 i b \sqrt {e} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 328, normalized size = 1.79 \begin {gather*} \frac {1}{4} \left (-\frac {4 b n}{d^2 x}-\frac {4 \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac {\sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b \sqrt {e} n \left (-\log (x)+\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {b \sqrt {e} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}-\frac {3 b \sqrt {e} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}+\frac {3 b \sqrt {e} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}\right ) \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 = 0.14, size = 933, normalized size = 5.10
method | result | size |
risch | \(-\frac {b e x \ln \left (x^{n}\right )}{2 d^{2} \left (e \,x^{2}+d \right )}-\frac {3 b e \arctan \left (\frac {x e}{\sqrt {e d}}\right ) \ln \left (x^{n}\right )}{2 d^{2} \sqrt {e d}}-\frac {a}{d^{2} x}-\frac {b \ln \left (x^{n}\right )}{d^{2} x}-\frac {3 a e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 d^{2} \sqrt {e d}}-\frac {a e x}{2 d^{2} \left (e \,x^{2}+d \right )}-\frac {3 b n e \dilog \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 d^{2} \sqrt {-e d}}+\frac {3 b n e \dilog \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 d^{2} \sqrt {-e d}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2} x}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2} x}-\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right ) x^{2}}{4 d^{2} \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {3 b e \arctan \left (\frac {x e}{\sqrt {e d}}\right ) n \ln \left (x \right )}{2 d^{2} \sqrt {e d}}-\frac {b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 d^{2} \sqrt {-e d}}+\frac {b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 d^{2} \sqrt {-e d}}+\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right ) x^{2}}{4 d^{2} \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 d^{2} x}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 d^{2} x}+\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 d^{2} \sqrt {e d}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {b \ln \left (c \right )}{d^{2} x}-\frac {b \ln \left (c \right ) e x}{2 d^{2} \left (e \,x^{2}+d \right )}+\frac {b n e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 d^{2} \sqrt {e d}}-\frac {b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 d \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 d \left (e \,x^{2}+d \right ) \sqrt {-e d}}-\frac {3 b \ln \left (c \right ) e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 d^{2} \sqrt {e d}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) e x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {b n}{d^{2} x}-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 d^{2} \sqrt {e d}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 d^{2} \sqrt {e d}}+\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) e \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 d^{2} \sqrt {e d}}\) | \(933\) |
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 {a + b \log {\left (c x^{n} \right )}}{x^{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 = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^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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