Optimal. Leaf size=135 \[ -\frac {b n}{4 e x^2}+\frac {b d n}{e^2 x}-\frac {a+b \log \left (c x^n\right )}{2 e x^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^3}-\frac {b d^2 n \text {Li}_2\left (-\frac {d x}{e}\right )}{e^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {269, 46, 2393,
2341, 2338, 2354, 2438} \begin {gather*} -\frac {b d^2 n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^3}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac {d^2 \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}-\frac {a+b \log \left (c x^n\right )}{2 e x^2}+\frac {b d n}{e^2 x}-\frac {b n}{4 e x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 2338
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e x^3}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x^2}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (e+d x)}\right ) \, dx\\ &=\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e^3}-\frac {d^3 \int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx}{e^3}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{e^2}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{e}\\ &=-\frac {b n}{4 e x^2}+\frac {b d n}{e^2 x}-\frac {a+b \log \left (c x^n\right )}{2 e x^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^3}+\frac {\left (b d^2 n\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{e^3}\\ &=-\frac {b n}{4 e x^2}+\frac {b d n}{e^2 x}-\frac {a+b \log \left (c x^n\right )}{2 e x^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^3}-\frac {b d^2 n \text {Li}_2\left (-\frac {d x}{e}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 124, normalized size = 0.92 \begin {gather*} -\frac {\frac {b e^2 n}{x^2}-\frac {4 b d e n}{x}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {4 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+4 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )+4 b d^2 n \text {Li}_2\left (-\frac {d x}{e}\right )}{4 e^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 = 0.06, size = 689, normalized size = 5.10
method | result | size |
risch | \(\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d}{2 e^{2} x}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \ln \left (d x +e \right )}{2 e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d}{2 e^{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 )}{4 e \,x^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \ln \left (x \right )}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \ln \left (d x +e \right )}{2 e^{3}}+\frac {b n \,d^{2} \dilog \left (-\frac {d x}{e}\right )}{e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{2} \ln \left (d x +e \right )}{e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \ln \left (x \right )}{2 e^{3}}-\frac {a}{2 e \,x^{2}}+\frac {b \ln \left (x^{n}\right ) d}{e^{2} x}+\frac {b \ln \left (x^{n}\right ) d^{2} \ln \left (x \right )}{e^{3}}+\frac {b d n}{e^{2} x}+\frac {b n \,d^{2} \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e^{3}}+\frac {a \,d^{2} \ln \left (x \right )}{e^{3}}+\frac {a d}{e^{2} x}-\frac {a \,d^{2} \ln \left (d x +e \right )}{e^{3}}-\frac {b n \,d^{2} \ln \left (x \right )^{2}}{2 e^{3}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 e \,x^{2}}-\frac {b \ln \left (x^{n}\right )}{2 e \,x^{2}}-\frac {b \ln \left (c \right )}{2 e \,x^{2}}+\frac {b \ln \left (c \right ) d^{2} \ln \left (x \right )}{e^{3}}+\frac {b \ln \left (c \right ) d}{e^{2} x}-\frac {b \ln \left (c \right ) d^{2} \ln \left (d x +e \right )}{e^{3}}-\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 ) d^{2} \ln \left (x \right )}{2 e^{3}}+\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 ) d^{2} \ln \left (d x +e \right )}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e \,x^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d}{2 e^{2} x}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2} \ln \left (x \right )}{2 e^{3}}-\frac {b n}{4 e \,x^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2} \ln \left (d x +e \right )}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e \,x^{2}}-\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 ) d}{2 e^{2} x}\) | \(689\) |
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 [A]
time = 63.20, size = 265, normalized size = 1.96 \begin {gather*} - \frac {a d^{3} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {a d^{2} \log {\left (x \right )}}{e^{3}} + \frac {a d}{e^{2} x} - \frac {a}{2 e x^{2}} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b d^{2} n \log {\left (x \right )}^{2}}{2 e^{3}} + \frac {b d^{2} \log {\left (x \right )} \log {\left (c x^{n} \right )}}{e^{3}} + \frac {b d n}{e^{2} x} + \frac {b d \log {\left (c x^{n} \right )}}{e^{2} x} - \frac {b n}{4 e x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 e x^{2}} \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^4\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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