Optimal. Leaf size=107 \[ -\frac {d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^2 p}+\frac {\left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^2 p} \]
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Rubi [A]
time = 0.10, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2504, 2446,
2436, 2337, 2209, 2437, 2347} \begin {gather*} \frac {\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^2 p}-\frac {d \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^2 p} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2446
Rule 2504
Rubi steps
\begin {align*} \int \frac {x^5}{\log \left (c \left (d+e x^3\right )^p\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} \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 )\\ &=\frac {\text {Subst}\left (\int \frac {d+e x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e}-\frac {d \text {Subst}\left (\int \frac {1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e}\\ &=\frac {\text {Subst}\left (\int \frac {x}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2}-\frac {d \text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2}\\ &=\frac {\left (\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^2 p}-\frac {\left (d \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^2 p}\\ &=-\frac {d \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^2 p}+\frac {\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^2 p}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 96, normalized size = 0.90 \begin {gather*} -\frac {\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-2/p} \left (d \left (c \left (d+e x^3\right )^p\right )^{\frac {1}{p}} \text {Ei}\left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )-\left (d+e x^3\right ) \text {Ei}\left (\frac {2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )\right )}{3 e^2 p} \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 = 0.66, size = 547, normalized size = 5.11
method | result | size |
risch | \(-\frac {\left (e \,x^{3}+d \right )^{2} c^{-\frac {2}{p}} \left (\left (e \,x^{3}+d \right )^{p}\right )^{-\frac {2}{p}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \left (-\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )+\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right )\right )}{p}} \expIntegral \left (1, -2 \ln \left (e \,x^{3}+d \right )-\frac {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 )+2 \ln \left (\left (e \,x^{3}+d \right )^{p}\right )-2 p \ln \left (e \,x^{3}+d \right )}{p}\right )}{3 e^{2} p}+\frac {d \left (e \,x^{3}+d \right ) c^{-\frac {1}{p}} \left (\left (e \,x^{3}+d \right )^{p}\right )^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \left (-\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )+\mathrm {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right )\right )}{2 p}} \expIntegral \left (1, -\ln \left (e \,x^{3}+d \right )-\frac {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 )+2 \ln \left (\left (e \,x^{3}+d \right )^{p}\right )-2 p \ln \left (e \,x^{3}+d \right )}{2 p}\right )}{3 e^{2} p}\) | \(547\) |
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 = 0.36, size = 68, normalized size = 0.64 \begin {gather*} -\frac {{\left (c^{\left (\frac {1}{p}\right )} d \operatorname {log\_integral}\left ({\left (x^{3} e + d\right )} c^{\left (\frac {1}{p}\right )}\right ) - \operatorname {log\_integral}\left ({\left (x^{6} e^{2} + 2 \, d x^{3} e + d^{2}\right )} c^{\frac {2}{p}}\right )\right )} e^{\left (-2\right )}}{3 \, c^{\frac {2}{p}} 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^{5}}{\log {\left (c \left (d + e x^{3}\right )^{p} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.65, size = 69, normalized size = 0.64 \begin {gather*} -\frac {d {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) e^{\left (-2\right )}}{3 \, c^{\left (\frac {1}{p}\right )} p} + \frac {{\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) e^{\left (-2\right )}}{3 \, c^{\frac {2}{p}} 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^5}{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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