Integrand size = 37, antiderivative size = 194 \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-\frac {p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{2 t u}+\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac {q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{2 t u}-p r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \]
[Out]
Time = 0.43 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2585, 2354, 2421, 6724, 2495} \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-p r \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {p r \log \left (\frac {b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}+p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )-q r \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {q r \log \left (\frac {d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}+q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \]
[In]
[Out]
Rule 2354
Rule 2421
Rule 2495
Rule 2585
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\log \left (i j^u (h x)^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = \text {Subst}\left (\text {Subst}\left (\int \frac {\log \left (h^{t u} i j^u x^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = \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}-\text {Subst}\left (\text {Subst}\left (\frac {(b p r) \int \frac {\log ^2\left (h^{t u} i j^u x^{t u}\right )}{a+b x} \, dx}{2 t u},h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )-\text {Subst}\left (\text {Subst}\left (\frac {(d q r) \int \frac {\log ^2\left (h^{t u} i j^u x^{t u}\right )}{c+d x} \, dx}{2 t u},h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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}+\text {Subst}\left (\text {Subst}\left ((p r) \int \frac {\log \left (h^{t u} i j^u x^{t u}\right ) \log \left (1+\frac {b x}{a}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )+\text {Subst}\left (\text {Subst}\left ((q r) \int \frac {\log \left (h^{t u} i j^u x^{t u}\right ) \log \left (1+\frac {d x}{c}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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 ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+\text {Subst}\left (\text {Subst}\left ((p r t u) \int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )+\text {Subst}\left (\text {Subst}\left ((q r t u) \int \frac {\text {Li}_2\left (-\frac {d x}{c}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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 ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+p r t u \text {Li}_3\left (-\frac {b x}{a}\right )+q r t u \text {Li}_3\left (-\frac {d x}{c}\right ) \\ \end{align*}
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 \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=p r t u \log (x) \log (h x) \log (a+b x)-p r t u \log ^2(h x) \log (a+b x)-p r \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+p r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+\frac {1}{2} p r t u \log ^2(h x) \log \left (1+\frac {b x}{a}\right )-p r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )+q r t u \log (x) \log (h x) \log (c+d x)-q r t u \log ^2(h x) \log (c+d x)-q r \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+q r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-t u \log (x) \log (h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{2} t u \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{2} q r t u \log ^2(h x) \log \left (1+\frac {d x}{c}\right )-q r \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )-p r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \]
[In]
[Out]
\[\int \frac {\ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right ) \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x}d x\]
[In]
[Out]
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}{x} \,d x \]
[In]
[Out]