Optimal. Leaf size=95 \[ -\frac {b n}{e x}-\frac {a+b \log \left (c x^n\right )}{e x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d n \text {Li}_2\left (-\frac {d x}{e}\right )}{e^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}+\frac {d \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {a+b \log \left (c x^n\right )}{e x}-\frac {b n}{e x} \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^3} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{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 )}{e^2 (e+d x)}\right ) \, dx\\ &=-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx}{e^2}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{e}\\ &=-\frac {b n}{e x}-\frac {a+b \log \left (c x^n\right )}{e x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^2}-\frac {(b d n) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac {b n}{e x}-\frac {a+b \log \left (c x^n\right )}{e x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d n \text {Li}_2\left (-\frac {d x}{e}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 88, normalized size = 0.93 \begin {gather*} -\frac {\frac {2 b e n}{x}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )-2 b d n \text {Li}_2\left (-\frac {d x}{e}\right )}{2 e^2} \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 = 504, normalized size = 5.31
method | result | size |
risch | \(-\frac {b \ln \left (x^{n}\right ) d \ln \left (x \right )}{e^{2}}+\frac {b n d \ln \left (x \right )^{2}}{2 e^{2}}-\frac {b n d \dilog \left (-\frac {d x}{e}\right )}{e^{2}}-\frac {b \ln \left (c \right )}{e x}-\frac {b n d \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \ln \left (d x +e \right )}{2 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e x}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e x}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \ln \left (x \right )}{2 e^{2}}-\frac {a}{e x}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 e x}-\frac {b \ln \left (c \right ) d \ln \left (x \right )}{e^{2}}+\frac {b \ln \left (c \right ) d \ln \left (d x +e \right )}{e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (d x +e \right )}{2 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (d x +e \right )}{2 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (x \right )}{2 e^{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 )}{2 e x}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (x \right )}{2 e^{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 \ln \left (x \right )}{2 e^{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 \ln \left (d x +e \right )}{2 e^{2}}-\frac {a d \ln \left (x \right )}{e^{2}}+\frac {a d \ln \left (d x +e \right )}{e^{2}}+\frac {b \ln \left (x^{n}\right ) d \ln \left (d x +e \right )}{e^{2}}-\frac {b \ln \left (x^{n}\right )}{e x}-\frac {b n}{e x}\) | \(504\) |
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 = 59.41, size = 216, normalized size = 2.27 \begin {gather*} \frac {a d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {a d \log {\left (x \right )}}{e^{2}} - \frac {a}{e x} - \frac {b d^{2} 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^{2}} + \frac {b d^{2} \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^{2}} + \frac {b d n \log {\left (x \right )}^{2}}{2 e^{2}} - \frac {b d \log {\left (x \right )} \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n}{e x} - \frac {b \log {\left (c x^{n} \right )}}{e x} \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^3\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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