Optimal. Leaf size=160 \[ a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {b d \text {PolyLog}\left (2,-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {PolyLog}\left (2,\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e m \text {PolyLog}\left (3,-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {PolyLog}\left (3,\frac {x^{-n}}{c}\right )}{2 n^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.41, antiderivative size = 160, 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,
6036, 6032, 6219, 6217, 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 (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \text {Li}_2\left (-\frac {x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}-\frac {b e \text {Li}_2\left (\frac {x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}+\frac {b e m \text {Li}_3\left (-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (\frac {x^{-n}}{c}\right )}{2 n^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2338
Rule 2421
Rule 6032
Rule 6036
Rule 6217
Rule 6219
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (a+b \coth ^{-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 \coth ^{-1}\left (c x^n\right )\right )}{x}+\frac {e \left (a+b \coth ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx\\ &=d \int \frac {a+b \coth ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac {\left (a+b \coth ^{-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 {\coth ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac {d \text {Subst}\left (\int \frac {a+b \coth ^{-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 (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1-\frac {x^{-n}}{c}\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1+\frac {x^{-n}}{c}\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 (-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {(b e m) \int \frac {\text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{x} \, dx}{2 n}+\frac {(b e m) \int \frac {\text {Li}_2\left (\frac {x^{-n}}{c}\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 (-\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-\frac {x^{-n}}{c}\right )}{2 n}-\frac {b d \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (\frac {x^{-n}}{c}\right )}{2 n}+\frac {b e m \text {Li}_3\left (-\frac {x^{-n}}{c}\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (\frac {x^{-n}}{c}\right )}{2 n^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.21, size = 131, normalized size = 0.82 \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} \left (a+b \coth ^{-1}\left (c x^n\right )-b \tanh ^{-1}\left (c x^n\right )\right ) \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.21, size = 920, normalized size = 5.75
method | result | size |
risch | \(\text {Expression too large to display}\) | \(920\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs.
\(2 (147) = 294\).
time = 0.41, size = 460, normalized size = 2.88 \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.
[In]
[Out]
________________________________________________________________________________________
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.
[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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acoth}\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.
[In]
[Out]
________________________________________________________________________________________