Optimal. Leaf size=136 \[ a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {PolyLog}\left (2,-c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,-c x^n\right )}{2 n}+\frac {b d \text {PolyLog}\left (2,c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,c x^n\right )}{2 n}+\frac {b e m \text {PolyLog}\left (3,-c x^n\right )}{2 n^2}-\frac {b e m \text {PolyLog}\left (3,c x^n\right )}{2 n^2} \]
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Rubi [A]
time = 0.39, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2338, 6874,
6035, 6031, 6218, 6216, 2421, 6724} \begin {gather*} a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}-\frac {b e \text {Li}_2\left (-c x^n\right ) \log \left (f x^m\right )}{2 n}+\frac {b e \text {Li}_2\left (c x^n\right ) \log \left (f x^m\right )}{2 n}+\frac {b e m \text {Li}_3\left (-c x^n\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (c x^n\right )}{2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2338
Rule 2421
Rule 6031
Rule 6035
Rule 6216
Rule 6218
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx &=\int \left (\frac {d \left (a+b \tanh ^{-1}\left (c x^n\right )\right )}{x}+\frac {e \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx\\ &=d \int \frac {a+b \tanh ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x} \, dx\\ &=(a e) \int \frac {\log \left (f x^m\right )}{x} \, dx+(b e) \int \frac {\tanh ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac {d \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}-\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1-c x^n\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1+c x^n\right )}{x} \, dx\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (c x^n\right )}{2 n}+\frac {(b e m) \int \frac {\text {Li}_2\left (-c x^n\right )}{x} \, dx}{2 n}-\frac {(b e m) \int \frac {\text {Li}_2\left (c x^n\right )}{x} \, dx}{2 n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e m \text {Li}_3\left (-c x^n\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (c x^n\right )}{2 n^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.19, size = 114, normalized size = 0.84 \begin {gather*} -\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n^2}+\frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right ) \left (d+e \log \left (f x^m\right )\right )}{n}+\frac {1}{2} a \log (x) \left (2 d-e m \log (x)+2 e \log \left (f x^m\right )\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 = 0.22, size = 668, normalized size = 4.91
method | result | size |
risch | \(-\frac {\ln \left (f \right ) \dilog \left (c \,x^{n}+1\right ) b e}{2 n}+\frac {i \pi \ln \left (x^{n}\right ) a e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{2 n}-\frac {i \pi \dilog \left (c \,x^{n}+1\right ) b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}-\frac {i \pi \dilog \left (c \,x^{n}+1\right ) b e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}+\frac {i \pi \dilog \left (1-c \,x^{n}\right ) b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}+\frac {i \pi \dilog \left (1-c \,x^{n}\right ) b e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}+\frac {b e m \polylog \left (3, -c \,x^{n}\right )}{2 n^{2}}-\frac {b e m \polylog \left (3, c \,x^{n}\right )}{2 n^{2}}+\frac {i \pi \ln \left (x^{n}\right ) a e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{2 n}-\frac {e b \dilog \left (c \,x^{n}+1\right ) \ln \left (x^{m}\right )}{2 n}-\frac {e b \dilog \left (c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}+\frac {\ln \left (f \right ) \dilog \left (1-c \,x^{n}\right ) b e}{2 n}-\frac {i \pi \dilog \left (1-c \,x^{n}\right ) b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{4 n}+\frac {\ln \left (f \right ) \ln \left (x^{n}\right ) a e}{n}+\frac {\ln \left (x^{n}\right ) a d}{n}+\frac {e b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right ) m \ln \left (x \right )}{2 n}+\frac {i \pi \dilog \left (c \,x^{n}+1\right ) b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{4 n}-\frac {i \pi \ln \left (x^{n}\right ) a e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{2 n}+\frac {e a \ln \left (x^{m}\right )^{2}}{2 m}+\frac {i \pi \dilog \left (c \,x^{n}+1\right ) b e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{4 n}-\frac {i \pi \dilog \left (1-c \,x^{n}\right ) b e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{4 n}-\frac {i \pi \ln \left (x^{n}\right ) a e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{2 n}-\frac {e b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}+\frac {e b m \ln \left (x \right ) \polylog \left (2, c \,x^{n}\right )}{2 n}+\frac {e b \dilog \left (c \,x^{n}\right ) m \ln \left (x \right )}{2 n}-\frac {e b m \ln \left (x \right ) \polylog \left (2, -c \,x^{n}\right )}{2 n}+\frac {e b \dilog \left (c \,x^{n}+1\right ) m \ln \left (x \right )}{2 n}+\frac {\dilog \left (1-c \,x^{n}\right ) b d}{2 n}-\frac {\dilog \left (c \,x^{n}+1\right ) b d}{2 n}\) | \(668\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (123) = 246\).
time = 0.35, size = 461, normalized size = 3.39 \begin {gather*} \frac {2 \, {\left (a m n^{2} \cosh \left (1\right ) + a m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n + {\left (b n \cosh \left (1\right ) + b n \sinh \left (1\right )\right )} \log \left (f\right ) + {\left (b m n \cosh \left (1\right ) + b m n \sinh \left (1\right )\right )} \log \left (x\right )\right )} {\rm Li}_2\left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, {\left (b d n + {\left (b n \cosh \left (1\right ) + b n \sinh \left (1\right )\right )} \log \left (f\right ) + {\left (b m n \cosh \left (1\right ) + b m n \sinh \left (1\right )\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right ) - {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1\right ) + {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right ) + 1\right ) + 4 \, {\left (a d n^{2} + {\left (a n^{2} \cosh \left (1\right ) + a n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right ) + {\left ({\left (b m n^{2} \cosh \left (1\right ) + b m n^{2} \sinh \left (1\right )\right )} \log \left (x\right )^{2} + 2 \, {\left (b d n^{2} + {\left (b n^{2} \cosh \left (1\right ) + b n^{2} \sinh \left (1\right )\right )} \log \left (f\right )\right )} \log \left (x\right )\right )} \log \left (-\frac {c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1}{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - 2 \, {\left (b m \cosh \left (1\right ) + b m \sinh \left (1\right )\right )} {\rm polylog}\left (3, c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) + 2 \, {\left (b m \cosh \left (1\right ) + b m \sinh \left (1\right )\right )} {\rm polylog}\left (3, -c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right )}{4 \, n^{2}} \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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