Optimal. Leaf size=86 \[ \frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}+\frac {2 e p^2 \text {Li}_2\left (1+\frac {e x^3}{d}\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2504, 2444,
2441, 2352} \begin {gather*} \frac {2 e p^2 \text {PolyLog}\left (2,\frac {e x^3}{d}+1\right )}{3 d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}+\frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2352
Rule 2441
Rule 2444
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}+\frac {(2 e p) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^3\right )}{3 d}\\ &=\frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}-\frac {\left (2 e^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^3\right )}{3 d}\\ &=\frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}+\frac {2 e p^2 \text {Li}_2\left (1+\frac {e x^3}{d}\right )}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 84, normalized size = 0.98 \begin {gather*} \frac {2 e p x^3 \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )-\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )+2 e p^2 x^3 \text {Li}_2\left (1+\frac {e x^3}{d}\right )}{3 d x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.48, size = 771, normalized size = 8.97
method | result | size |
risch | \(-\frac {\ln \left (\left (e \,x^{3}+d \right )^{p}\right )^{2}}{3 x^{3}}+\frac {2 p e \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \ln \left (x \right )}{d}-\frac {2 p e \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \ln \left (e \,x^{3}+d \right )}{3 d}-\frac {2 p^{2} e \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3} e +d \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{d}+\frac {p^{2} e \ln \left (e \,x^{3}+d \right )^{2}}{3 d}+\frac {i p e \ln \left (e \,x^{3}+d \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{3 d}+\frac {i p e \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{d}+\frac {i \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{3}}{3 x^{3}}-\frac {i p e \ln \left (e \,x^{3}+d \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{3 d}-\frac {2 \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \ln \left (c \right )}{3 x^{3}}-\frac {i p e \ln \left (e \,x^{3}+d \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{3 d}+\frac {i \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{3 x^{3}}-\frac {i p e \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{d}+\frac {i p e \ln \left (e \,x^{3}+d \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{3}}{3 d}+\frac {2 p e \ln \left (x \right ) \ln \left (c \right )}{d}-\frac {i p e \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{3}}{d}-\frac {i \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{3 x^{3}}+\frac {i p e \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{d}-\frac {i \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{3 x^{3}}-\frac {2 p e \ln \left (e \,x^{3}+d \right ) \ln \left (c \right )}{3 d}-\frac {\left (i \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )\right )^{2}}{12 x^{3}}\) | \(771\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 118, normalized size = 1.37 \begin {gather*} \frac {1}{3} \, e^{2} p^{2} {\left (\frac {\log \left (e x^{3} + d\right )^{2}}{d e} - \frac {2 \, {\left (3 \, \log \left (\frac {e x^{3}}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x^{3}}{d}\right )\right )}}{d e}\right )} - \frac {2}{3} \, e p {\left (\frac {\log \left (e x^{3} + d\right )}{d} - \frac {\log \left (x^{3}\right )}{d}\right )} \log \left ({\left (e x^{3} + d\right )}^{p} c\right ) - \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{3 \, x^{3}} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{4}}\, dx \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 {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________