Integrand size = 10, antiderivative size = 81 \[ \int \log ^4(c (d+e x)) \, dx=24 x-\frac {24 (d+e x) \log (c (d+e x))}{e}+\frac {12 (d+e x) \log ^2(c (d+e x))}{e}-\frac {4 (d+e x) \log ^3(c (d+e x))}{e}+\frac {(d+e x) \log ^4(c (d+e x))}{e} \]
24*x-24*(e*x+d)*ln(c*(e*x+d))/e+12*(e*x+d)*ln(c*(e*x+d))^2/e-4*(e*x+d)*ln( c*(e*x+d))^3/e+(e*x+d)*ln(c*(e*x+d))^4/e
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \log ^4(c (d+e x)) \, dx=\frac {24 e x-24 (d+e x) \log (c (d+e x))+12 (d+e x) \log ^2(c (d+e x))-4 (d+e x) \log ^3(c (d+e x))+(d+e x) \log ^4(c (d+e x))}{e} \]
(24*e*x - 24*(d + e*x)*Log[c*(d + e*x)] + 12*(d + e*x)*Log[c*(d + e*x)]^2 - 4*(d + e*x)*Log[c*(d + e*x)]^3 + (d + e*x)*Log[c*(d + e*x)]^4)/e
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2836, 2733, 2733, 2733, 2732}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \log ^4(c (d+e x)) \, dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \log ^4(c (d+e x))d(d+e x)}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \log ^4(c (d+e x))-4 \int \log ^3(c (d+e x))d(d+e x)}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \log ^4(c (d+e x))-4 \left ((d+e x) \log ^3(c (d+e x))-3 \int \log ^2(c (d+e x))d(d+e x)\right )}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \log ^4(c (d+e x))-4 \left ((d+e x) \log ^3(c (d+e x))-3 \left ((d+e x) \log ^2(c (d+e x))-2 \int \log (c (d+e x))d(d+e x)\right )\right )}{e}\) |
\(\Big \downarrow \) 2732 |
\(\displaystyle \frac {(d+e x) \log ^4(c (d+e x))-4 \left ((d+e x) \log ^3(c (d+e x))-3 \left ((d+e x) \log ^2(c (d+e x))-2 ((d+e x) \log (c (d+e x))-d-e x)\right )\right )}{e}\) |
((d + e*x)*Log[c*(d + e*x)]^4 - 4*((d + e*x)*Log[c*(d + e*x)]^3 - 3*((d + e*x)*Log[c*(d + e*x)]^2 - 2*(-d - e*x + (d + e*x)*Log[c*(d + e*x)]))))/e
3.1.1.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x ] /; FreeQ[{c, n}, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {\left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{4}}{e}-\frac {4 \left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{3}}{e}+\frac {12 \left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{2}}{e}-24 x \ln \left (c \left (e x +d \right )\right )+24 x -\frac {24 d \ln \left (e x +d \right )}{e}\) | \(87\) |
derivativedivides | \(\frac {\ln \left (c e x +c d \right )^{4} \left (c e x +c d \right )-4 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}+12 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}-24 \left (c e x +c d \right ) \ln \left (c e x +c d \right )+24 c e x +24 c d}{c e}\) | \(99\) |
default | \(\frac {\ln \left (c e x +c d \right )^{4} \left (c e x +c d \right )-4 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}+12 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}-24 \left (c e x +c d \right ) \ln \left (c e x +c d \right )+24 c e x +24 c d}{c e}\) | \(99\) |
norman | \(x \ln \left (c \left (e x +d \right )\right )^{4}+\frac {d \ln \left (c \left (e x +d \right )\right )^{4}}{e}+24 x -24 x \ln \left (c \left (e x +d \right )\right )+12 x \ln \left (c \left (e x +d \right )\right )^{2}-4 x \ln \left (c \left (e x +d \right )\right )^{3}-\frac {24 d \ln \left (c \left (e x +d \right )\right )}{e}+\frac {12 d \ln \left (c \left (e x +d \right )\right )^{2}}{e}-\frac {4 d \ln \left (c \left (e x +d \right )\right )^{3}}{e}\) | \(115\) |
parallelrisch | \(\frac {x \ln \left (c \left (e x +d \right )\right )^{4} e -4 x \ln \left (c \left (e x +d \right )\right )^{3} e +\ln \left (c \left (e x +d \right )\right )^{4} d +12 x \ln \left (c \left (e x +d \right )\right )^{2} e -4 \ln \left (c \left (e x +d \right )\right )^{3} d -24 \ln \left (c \left (e x +d \right )\right ) x e +12 \ln \left (c \left (e x +d \right )\right )^{2} d +24 e x -24 d \ln \left (c \left (e x +d \right )\right )-24 d}{e}\) | \(115\) |
(e*x+d)*ln(c*(e*x+d))^4/e-4*(e*x+d)*ln(c*(e*x+d))^3/e+12*(e*x+d)*ln(c*(e*x +d))^2/e-24*x*ln(c*(e*x+d))+24*x-24*d/e*ln(e*x+d)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \log ^4(c (d+e x)) \, dx=\frac {{\left (e x + d\right )} \log \left (c e x + c d\right )^{4} - 4 \, {\left (e x + d\right )} \log \left (c e x + c d\right )^{3} + 12 \, {\left (e x + d\right )} \log \left (c e x + c d\right )^{2} + 24 \, e x - 24 \, {\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \]
((e*x + d)*log(c*e*x + c*d)^4 - 4*(e*x + d)*log(c*e*x + c*d)^3 + 12*(e*x + d)*log(c*e*x + c*d)^2 + 24*e*x - 24*(e*x + d)*log(c*e*x + c*d))/e
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int \log ^4(c (d+e x)) \, dx=24 e \left (- \frac {d \log {\left (d + e x \right )}}{e^{2}} + \frac {x}{e}\right ) - 24 x \log {\left (c \left (d + e x\right ) \right )} + \frac {\left (- 4 d - 4 e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{3}}{e} + \frac {\left (d + e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{4}}{e} + \frac {\left (12 d + 12 e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{2}}{e} \]
24*e*(-d*log(d + e*x)/e**2 + x/e) - 24*x*log(c*(d + e*x)) + (-4*d - 4*e*x) *log(c*(d + e*x))**3/e + (d + e*x)*log(c*(d + e*x))**4/e + (12*d + 12*e*x) *log(c*(d + e*x))**2/e
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (81) = 162\).
Time = 0.20 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.32 \[ \int \log ^4(c (d+e x)) \, dx=-4 \, e {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )} c\right )^{3} + x \log \left ({\left (e x + d\right )} c\right )^{4} + {\left (e {\left (\frac {4 \, {\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{3}} - \frac {d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )}{e^{3}}\right )} - \frac {6 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} \log \left ({\left (e x + d\right )} c\right )^{2}}{e^{2}}\right )} e \]
-4*e*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)*c)^3 + x*log((e*x + d)*c)^4 + (e*(4*(d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x + 6*d*log(e*x + d)) *log((e*x + d)*c)/e^3 - (d*log(e*x + d)^4 + 4*d*log(e*x + d)^3 + 12*d*log( e*x + d)^2 - 24*e*x + 24*d*log(e*x + d))/e^3) - 6*(d*log(e*x + d)^2 - 2*e* x + 2*d*log(e*x + d))*log((e*x + d)*c)^2/e^2)*e
Time = 0.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int \log ^4(c (d+e x)) \, dx=\frac {{\left (e x + d\right )} \log \left ({\left (e x + d\right )} c\right )^{4}}{e} - \frac {4 \, {\left (e x + d\right )} \log \left ({\left (e x + d\right )} c\right )^{3}}{e} + \frac {12 \, {\left (e x + d\right )} \log \left ({\left (e x + d\right )} c\right )^{2}}{e} - \frac {24 \, {\left (e x + d\right )} \log \left ({\left (e x + d\right )} c\right )}{e} + \frac {24 \, {\left (e x + d\right )}}{e} \]
(e*x + d)*log((e*x + d)*c)^4/e - 4*(e*x + d)*log((e*x + d)*c)^3/e + 12*(e* x + d)*log((e*x + d)*c)^2/e - 24*(e*x + d)*log((e*x + d)*c)/e + 24*(e*x + d)/e
Time = 1.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.47 \[ \int \log ^4(c (d+e x)) \, dx=24\,x-24\,x\,\ln \left (c\,d+c\,e\,x\right )+12\,x\,{\ln \left (c\,d+c\,e\,x\right )}^2-4\,x\,{\ln \left (c\,d+c\,e\,x\right )}^3+x\,{\ln \left (c\,d+c\,e\,x\right )}^4+\frac {12\,d\,{\ln \left (c\,d+c\,e\,x\right )}^2}{e}-\frac {4\,d\,{\ln \left (c\,d+c\,e\,x\right )}^3}{e}+\frac {d\,{\ln \left (c\,d+c\,e\,x\right )}^4}{e}-\frac {24\,d\,\ln \left (d+e\,x\right )}{e} \]