Integrand size = 37, antiderivative size = 194 \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-\frac {p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{2 t u}+\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac {q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{2 t u}-p r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \]
-1/2*p*r*ln(i*(j*(h*x)^t)^u)^2*ln(1+b*x/a)/t/u+1/2*ln(i*(j*(h*x)^t)^u)^2*l n(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/t/u-1/2*q*r*ln(i*(j*(h*x)^t)^u)^2*ln(1+d*x/ c)/t/u-p*r*ln(i*(j*(h*x)^t)^u)*polylog(2,-b*x/a)-q*r*ln(i*(j*(h*x)^t)^u)*p olylog(2,-d*x/c)+p*r*t*u*polylog(3,-b*x/a)+q*r*t*u*polylog(3,-d*x/c)
Leaf count is larger than twice the leaf count of optimal. \(451\) vs. \(2(194)=388\).
Time = 0.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.32 \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=p r t u \log (x) \log (h x) \log (a+b x)-p r t u \log ^2(h x) \log (a+b x)-p r \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+p r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+\frac {1}{2} p r t u \log ^2(h x) \log \left (1+\frac {b x}{a}\right )-p r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )+q r t u \log (x) \log (h x) \log (c+d x)-q r t u \log ^2(h x) \log (c+d x)-q r \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+q r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-t u \log (x) \log (h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{2} t u \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{2} q r t u \log ^2(h x) \log \left (1+\frac {d x}{c}\right )-q r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )-p r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \]
p*r*t*u*Log[x]*Log[h*x]*Log[a + b*x] - p*r*t*u*Log[h*x]^2*Log[a + b*x] - p *r*Log[x]*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] + p*r*Log[h*x]*Log[i*(j*(h*x)^ t)^u]*Log[a + b*x] + (p*r*t*u*Log[h*x]^2*Log[1 + (b*x)/a])/2 - p*r*Log[h*x ]*Log[i*(j*(h*x)^t)^u]*Log[1 + (b*x)/a] + q*r*t*u*Log[x]*Log[h*x]*Log[c + d*x] - q*r*t*u*Log[h*x]^2*Log[c + d*x] - q*r*Log[x]*Log[i*(j*(h*x)^t)^u]*L og[c + d*x] + q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[c + d*x] - t*u*Log[x]* Log[h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (t*u*Log[h*x]^2*Log[e*(f*( a + b*x)^p*(c + d*x)^q)^r])/2 + Log[x]*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (q*r*t*u*Log[h*x]^2*Log[1 + (d*x)/c])/2 - q*r*Log [h*x]*Log[i*(j*(h*x)^t)^u]*Log[1 + (d*x)/c] - p*r*Log[i*(j*(h*x)^t)^u]*Pol yLog[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)] + p* r*t*u*PolyLog[3, -((b*x)/a)] + q*r*t*u*PolyLog[3, -((d*x)/c)]
Time = 1.16 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2895, 2895, 2985, 2754, 2821, 7143}
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.
| \(\displaystyle \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x}dx\) |
| \(\Big \downarrow \) 2985 |
| \(\displaystyle -\frac {b p r \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right )}{a+b x}dx}{2 t u}-\frac {d q r \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right )}{c+d x}dx}{2 t u}+\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {2 t u \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {b x}{a}+1\right )}{x}dx}{b}\right )}{2 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {2 t u \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {d x}{c}+1\right )}{x}dx}{d}\right )}{2 t u}+\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {2 t u \left (t u \int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{2 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {2 t u \left (t u \int \frac {\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{2 t u}+\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {2 t u \left (t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{2 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {2 t u \left (t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{2 t u}\) |
(Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*t*u) - (b *p*r*((Log[i*(j*(h*x)^t)^u]^2*Log[1 + (b*x)/a])/b - (2*t*u*(-(Log[i*(j*(h* x)^t)^u]*PolyLog[2, -((b*x)/a)]) + t*u*PolyLog[3, -((b*x)/a)]))/b))/(2*t*u ) - (d*q*r*((Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c])/d - (2*t*u*(-(Log[i* (j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)]) + t*u*PolyLog[3, -((d*x)/c)]))/d))/ (2*t*u)
3.1.58.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.) )^(r_.)]*((s_.) + Log[(i_.)*((g_.) + (h_.)*(x_))^(n_.)]*(t_.))^(m_.))/((j_. ) + (k_.)*(x_)), x_Symbol] :> Simp[(s + t*Log[i*(g + h*x)^n])^(m + 1)*(Log[ e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*n*t*(m + 1))), x] + (-Simp[b*p*(r/(k*n* t*(m + 1))) Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Si mp[d*q*(r/(k*n*t*(m + 1))) Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(c + d* x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r} , x] && NeQ[b*c - a*d, 0] && EqQ[h*j - g*k, 0] && IGtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right ) \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x}d x\]
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
-1/2*(t*u*log(x)^2 - 2*(t*u*log(h) + u*log(j) + log(i))*log(x) - 2*log(x)* log((x^t)^u))*log(((b*x + a)^p)^r) - 1/2*(t*u*log(x)^2 - 2*(t*u*log(h) + u *log(j) + log(i))*log(x) - 2*log(x)*log((x^t)^u))*log(((d*x + c)^q)^r) - i ntegrate(-1/2*(2*((t*u*log(h) + u*log(j) + log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log(f))*b*d*x^2 + 2*((t*u*log(h) + u*log(j) + log (i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log(f))*a*c + ((p*r*t *u + q*r*t*u)*b*d*x^2 + (b*c*p*r*t*u + a*d*q*r*t*u)*x)*log(x)^2 + 2*(((t*u *log(h) + u*log(j) + log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i ))*log(f))*b*c + ((t*u*log(h) + u*log(j) + log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log(f))*a*d)*x + 2*((r*log(f) + log(e))*b*d*x^2 + (r*log(f) + log(e))*a*c + ((r*log(f) + log(e))*b*c + (r*log(f) + log(e))* a*d)*x - ((p*r + q*r)*b*d*x^2 + (b*c*p*r + a*d*q*r)*x)*log(x))*log((x^t)^u ) - 2*(((p*r*t*u + q*r*t*u)*log(h) + (p*r + q*r)*log(i) + (p*r*u + q*r*u)* log(j))*b*d*x^2 + ((p*r*t*u*log(h) + p*r*u*log(j) + p*r*log(i))*b*c + (q*r *t*u*log(h) + q*r*u*log(j) + q*r*log(i))*a*d)*x)*log(x))/(b*d*x^3 + a*c*x + (b*c + a*d)*x^2), x)
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}{x} \,d x \]