Optimal. Leaf size=135 \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2334,
2337, 2209} \begin {gather*} \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2209
Rule 2334
Rule 2337
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx,x,d+e x\right )}{e}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e n}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e n^2}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e n^3}\\ &=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 144, normalized size = 1.07 \begin {gather*} -\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (-\text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {a}{b n}} n \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (a+b n+b \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.70, size = 734, normalized size = 5.44
method | result | size |
risch | \(-\frac {2 b e n x +2 b d n -i \pi b d \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b d \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b e x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b e x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+2 \ln \left (c \right ) b e x +2 b e x \ln \left (\left (e x +d \right )^{n}\right )+2 d b \ln \left (c \right )+2 a e x +2 b d \ln \left (\left (e x +d \right )^{n}\right )+2 a d}{\left (2 a +2 b \ln \left (c \right )+2 b \ln \left (\left (e x +d \right )^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}\right )^{2} b^{2} n^{2} e}-\frac {\left (e x +d \right ) \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 a}{2 b n}} \expIntegral \left (1, -\ln \left (e x +d \right )-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (\left (e x +d \right )^{n}\right )-n \ln \left (e x +d \right )\right )+2 a}{2 b n}\right )}{2 b^{3} n^{3} e}\) | \(734\) |
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. 277 vs.
\(2 (132) = 264\).
time = 0.38, size = 277, normalized size = 2.05 \begin {gather*} -\frac {{\left ({\left (b^{2} d n^{2} + a b d n + {\left (b^{2} n^{2} + a b n\right )} x e + {\left (b^{2} n^{2} x e + b^{2} d n^{2}\right )} \log \left (x e + d\right ) + {\left (b^{2} n x e + b^{2} d n\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \left (x e + d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x e + d\right )\right )} \operatorname {log\_integral}\left ({\left (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e + 2 \, {\left (b^{5} n^{4} e \log \left (c\right ) + a b^{4} n^{4} e\right )} \log \left (x e + d\right )\right )}} \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 {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1322 vs.
\(2 (132) = 264\).
time = 3.33, size = 1322, normalized size = 9.79 \begin {gather*} \frac {b^{2} n^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x e + d\right )^{2}}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} b^{2} n^{2} \log \left (x e + d\right )}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )}} + \frac {b^{2} n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x e + d\right ) \log \left (c\right )}{{\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} b^{2} n^{2}}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )}} + \frac {a b n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x e + d\right )}{{\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} b^{2} n \log \left (c\right )}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )}} + \frac {b^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (c\right )^{2}}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (x e + d\right )} a b n}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )}} + \frac {a b {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (c\right )}{{\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{2 \, {\left (b^{5} n^{5} e \log \left (x e + d\right )^{2} + 2 \, b^{5} n^{4} e \log \left (x e + d\right ) \log \left (c\right ) + 2 \, a b^{4} n^{4} e \log \left (x e + d\right ) + b^{5} n^{3} e \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} e \log \left (c\right ) + a^{2} b^{3} n^{3} e\right )} c^{\left (\frac {1}{n}\right )}} \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 {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________