Optimal. Leaf size=195 \[ \frac {d^2 \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^3 p^2}-\frac {4 d \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^3 p^2}+\frac {\left (d+e x^3\right )^3 \left (c \left (d+e x^3\right )^p\right )^{-3/p} \text {Ei}\left (\frac {3 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{e^3 p^2}-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \]
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Rubi [A]
time = 0.25, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2504, 2447,
2446, 2436, 2337, 2209, 2437, 2347} \begin {gather*} \frac {d^2 \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^3 p^2}+\frac {\left (d+e x^3\right )^3 \left (c \left (d+e x^3\right )^p\right )^{-3/p} \text {Ei}\left (\frac {3 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{e^3 p^2}-\frac {4 d \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^3 p^2}-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2504
Rubi steps
\begin {align*} \int \frac {x^8}{\log ^2\left (c \left (d+e x^3\right )^p\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\log ^2\left (c (d+e x)^p\right )} \, dx,x,x^3\right )\\ &=-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac {\text {Subst}\left (\int \frac {x^2}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{p}+\frac {(2 d) \text {Subst}\left (\int \frac {x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e p}\\ &=-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac {\text {Subst}\left (\int \left (\frac {d^2}{e^2 \log \left (c (d+e x)^p\right )}-\frac {2 d (d+e x)}{e^2 \log \left (c (d+e x)^p\right )}+\frac {(d+e x)^2}{e^2 \log \left (c (d+e x)^p\right )}\right ) \, dx,x,x^3\right )}{p}+\frac {(2 d) \text {Subst}\left (\int \left (-\frac {d}{e \log \left (c (d+e x)^p\right )}+\frac {d+e x}{e \log \left (c (d+e x)^p\right )}\right ) \, dx,x,x^3\right )}{3 e p}\\ &=-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac {\text {Subst}\left (\int \frac {(d+e x)^2}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{e^2 p}+\frac {(2 d) \text {Subst}\left (\int \frac {d+e x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e^2 p}-\frac {(2 d) \text {Subst}\left (\int \frac {d+e x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{e^2 p}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e^2 p}+\frac {d^2 \text {Subst}\left (\int \frac {1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{e^2 p}\\ &=-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac {\text {Subst}\left (\int \frac {x^2}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{e^3 p}+\frac {(2 d) \text {Subst}\left (\int \frac {x}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^3 p}-\frac {(2 d) \text {Subst}\left (\int \frac {x}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{e^3 p}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^3 p}+\frac {d^2 \text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{e^3 p}\\ &=-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}+\frac {\left (\left (d+e x^3\right )^3 \left (c \left (d+e x^3\right )^p\right )^{-3/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{e^3 p^2}+\frac {\left (2 d \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^3 p^2}-\frac {\left (2 d \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{e^3 p^2}-\frac {\left (2 d^2 \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^3 p^2}+\frac {\left (d^2 \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{e^3 p^2}\\ &=\frac {d^2 \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^3 p^2}-\frac {4 d \left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^3 p^2}+\frac {\left (d+e x^3\right )^3 \left (c \left (d+e x^3\right )^p\right )^{-3/p} \text {Ei}\left (\frac {3 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{e^3 p^2}-\frac {x^6 \left (d+e x^3\right )}{3 e p \log \left (c \left (d+e x^3\right )^p\right )}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 290, normalized size = 1.49 \begin {gather*} \frac {\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-3/p} \left (-e^2 p x^6 \left (c \left (d+e x^3\right )^p\right )^{3/p}+d^2 \left (c \left (d+e x^3\right )^p\right )^{2/p} \text {Ei}\left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{p}\right ) \log \left (c \left (d+e x^3\right )^p\right )-4 d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{\frac {1}{p}} \text {Ei}\left (\frac {2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right ) \log \left (c \left (d+e x^3\right )^p\right )+3 d^2 \text {Ei}\left (\frac {3 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right ) \log \left (c \left (d+e x^3\right )^p\right )+6 d e x^3 \text {Ei}\left (\frac {3 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right ) \log \left (c \left (d+e x^3\right )^p\right )+3 e^2 x^6 \text {Ei}\left (\frac {3 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right ) \log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^3 p^2 \log \left (c \left (d+e x^3\right )^p\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 = 0.80, size = 2564, normalized size = 13.15
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2564\) |
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 [A]
time = 0.38, size = 211, normalized size = 1.08 \begin {gather*} -\frac {4 \, {\left (d p \log \left (x^{3} e + d\right ) + d \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (x^{6} e^{2} + 2 \, d x^{3} e + d^{2}\right )} c^{\frac {2}{p}}\right ) - {\left (d^{2} p \log \left (x^{3} e + d\right ) + d^{2} \log \left (c\right )\right )} c^{\frac {2}{p}} \operatorname {log\_integral}\left ({\left (x^{3} e + d\right )} c^{\left (\frac {1}{p}\right )}\right ) + {\left (p x^{9} e^{3} + d p x^{6} e^{2}\right )} c^{\frac {3}{p}} - 3 \, {\left (p \log \left (x^{3} e + d\right ) + \log \left (c\right )\right )} \operatorname {log\_integral}\left ({\left (x^{9} e^{3} + 3 \, d x^{6} e^{2} + 3 \, d^{2} x^{3} e + d^{3}\right )} c^{\frac {3}{p}}\right )}{3 \, {\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\frac {3}{p}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 490 vs.
\(2 (200) = 400\).
time = 5.35, size = 490, normalized size = 2.51 \begin {gather*} -\frac {1}{3} \, d^{2} {\left (\frac {{\left (x^{3} e + d\right )} p}{p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )} - \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) \log \left (x^{3} e + d\right )}{{\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} - \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{{\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}}\right )} - \frac {{\left (x^{3} e + d\right )}^{3} p}{3 \, {\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )}} + \frac {2 \, {\left (x^{3} e + d\right )}^{2} d p}{3 \, {\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )}} - \frac {4 \, d p {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) \log \left (x^{3} e + d\right )}{3 \, {\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\frac {2}{p}}} + \frac {p {\rm Ei}\left (\frac {3 \, \log \left (c\right )}{p} + 3 \, \log \left (x^{3} e + d\right )\right ) \log \left (x^{3} e + d\right )}{{\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\frac {3}{p}}} - \frac {4 \, d {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{3 \, {\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\frac {2}{p}}} + \frac {{\rm Ei}\left (\frac {3 \, \log \left (c\right )}{p} + 3 \, \log \left (x^{3} e + d\right )\right ) \log \left (c\right )}{{\left (p^{3} e^{3} \log \left (x^{3} e + d\right ) + p^{2} e^{3} \log \left (c\right )\right )} c^{\frac {3}{p}}} \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 {x^8}{{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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