Optimal. Leaf size=194 \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\frac{\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.603361, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {2499, 2317, 2374, 6589, 2445} \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\frac{\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )}{2 t u} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2499
Rule 2317
Rule 2374
Rule 6589
Rule 2445
Rubi steps
\begin{align*} \int \frac{\log \left (58 \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 &=\operatorname{Subst}\left (\int \frac{\log \left (58 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,58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=\operatorname{Subst}\left (\operatorname{Subst}\left (\int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=\frac{\log ^2\left (58 \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}-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(b p r) \int \frac{\log ^2\left (58 h^{t u} j^u x^{t u}\right )}{a+b x} \, dx}{2 t u},58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(d q r) \int \frac{\log ^2\left (58 h^{t u} j^u x^{t u}\right )}{c+d x} \, dx}{2 t u},58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \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 (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r) \int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{b x}{a}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r) \int \frac{\log \left (58 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{d x}{c}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \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 (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}-p r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r t u) \int \frac{\text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r t u) \int \frac{\text{Li}_2\left (-\frac{d x}{c}\right )}{x} \, dx,58 h^{t u} j^u x^{t u},58 j^u (h x)^{t u}\right ),58 j^u (h x)^{t u},58 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^2\left (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{2 t u}+\frac{\log ^2\left (58 \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 (58 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{2 t u}-p r \log \left (58 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log \left (58 \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*}
Mathematica [B] time = 0.448249, size = 451, normalized size = 2.32 \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )+q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right )+\log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac{1}{2} t u \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-t u \log (x) \log (h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (h x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (h x) \log \left (\frac{b x}{a}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-p r \log (x) \log (a+b x) \log \left (i \left (j (h x)^t\right )^u\right )-p r t u \log ^2(h x) \log (a+b x)+\frac{1}{2} p r t u \log ^2(h x) \log \left (\frac{b x}{a}+1\right )+p r t u \log (x) \log (h x) \log (a+b x)+q r \log (h x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (h x) \log \left (\frac{d x}{c}+1\right ) \log \left (i \left (j (h x)^t\right )^u\right )-q r \log (x) \log (c+d x) \log \left (i \left (j (h x)^t\right )^u\right )-q r t u \log ^2(h x) \log (c+d x)+\frac{1}{2} q r t u \log ^2(h x) \log \left (\frac{d x}{c}+1\right )+q r t u \log (x) \log (h x) \log (c+d x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.784, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( i \left ( j \left ( hx \right ) ^{t} \right ) ^{u} \right ) \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]