Optimal. Leaf size=44 \[ -\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \text {Li}_2\left (-\frac {e}{d x}\right )}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2375, 2438}
\begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {e}{d x}\right )}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2375
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx &=-\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {(b n) \int \frac {\log \left (1+\frac {e}{d x}\right )}{x} \, dx}{e}\\ &=-\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \text {Li}_2\left (-\frac {e}{d x}\right )}{e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 63, normalized size = 1.43 \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {d x}{e}\right )\right )}{2 b e n}-\frac {b n \text {Li}_2\left (-\frac {d x}{e}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.06, size = 336, normalized size = 7.64
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{e}-\frac {b \ln \left (x^{n}\right ) \ln \left (d x +e \right )}{e}-\frac {b n \ln \left (x \right )^{2}}{2 e}+\frac {b n \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e}+\frac {b n \dilog \left (-\frac {d x}{e}\right )}{e}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d x +e \right )}{2 e}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (d x +e \right )}{2 e}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x \right )}{2 e}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x \right )}{2 e}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x \right )}{2 e}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x \right )}{2 e}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 e}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 e}+\frac {b \ln \left (c \right ) \ln \left (x \right )}{e}-\frac {b \ln \left (c \right ) \ln \left (d x +e \right )}{e}+\frac {a \ln \left (x \right )}{e}-\frac {a \ln \left (d x +e \right )}{e}\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 7.47, size = 173, normalized size = 3.93 \begin {gather*} \frac {2 a d \left (\begin {cases} - \frac {x}{e} - \frac {1}{2 d} & \text {for}\: d = 0 \\\frac {\log {\left (2 d x \right )}}{2 d} & \text {otherwise} \end {cases}\right )}{e} - \frac {2 a d \left (\begin {cases} \frac {x}{e} + \frac {1}{2 d} & \text {for}\: d = 0 \\\frac {\log {\left (2 d x + 2 e \right )}}{2 d} & \text {otherwise} \end {cases}\right )}{e} + b n \left (\begin {cases} - \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\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 (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{x} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________