Optimal. Leaf size=262 \[ \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-p r \text {Li}_2\left (-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 p r t u \text {Li}_3\left (-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {p r \log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-2 p r t^2 u^2 \text {Li}_4\left (-\frac {b x}{a}\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 q r t u \text {Li}_3\left (-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {q r \log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-2 q r t^2 u^2 \text {Li}_4\left (-\frac {d x}{c}\right ) \]
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Rubi [A] time = 0.91, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2499, 2317, 2374, 2383, 6589, 2445} \[ -p r \text {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 p r t u \text {PolyLog}\left (3,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 p r t^2 u^2 \text {PolyLog}\left (4,-\frac {b x}{a}\right )-q r \text {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 q r t u \text {PolyLog}\left (3,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 q r t^2 u^2 \text {PolyLog}\left (4,-\frac {d x}{c}\right )+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {p r \log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-\frac {q r \log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u} \]
Antiderivative was successfully verified.
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Rule 2317
Rule 2374
Rule 2383
Rule 2445
Rule 2499
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^2\left (57 \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 ^2\left (57 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,57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=\operatorname {Subst}\left (\operatorname {Subst}\left (\int \frac {\log ^2\left (57 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,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=\frac {\log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\operatorname {Subst}\left (\operatorname {Subst}\left (\frac {(b p r) \int \frac {\log ^3\left (57 h^{t u} j^u x^{t u}\right )}{a+b x} \, dx}{3 t u},57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )-\operatorname {Subst}\left (\operatorname {Subst}\left (\frac {(d q r) \int \frac {\log ^3\left (57 h^{t u} j^u x^{t u}\right )}{c+d x} \, dx}{3 t u},57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac {p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}+\operatorname {Subst}\left (\operatorname {Subst}\left ((p r) \int \frac {\log ^2\left (57 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac {b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )+\operatorname {Subst}\left (\operatorname {Subst}\left ((q r) \int \frac {\log ^2\left (57 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac {d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac {p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+\operatorname {Subst}\left (\operatorname {Subst}\left ((2 p r t u) \int \frac {\log \left (57 h^{t u} j^u x^{t u}\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )+\operatorname {Subst}\left (\operatorname {Subst}\left ((2 q r t u) \int \frac {\log \left (57 h^{t u} j^u x^{t u}\right ) \text {Li}_2\left (-\frac {d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac {p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {d x}{c}\right )-\operatorname {Subst}\left (\operatorname {Subst}\left (\left (2 p r t^2 u^2\right ) \int \frac {\text {Li}_3\left (-\frac {b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )-\operatorname {Subst}\left (\operatorname {Subst}\left (\left (2 q r t^2 u^2\right ) \int \frac {\text {Li}_3\left (-\frac {d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac {p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {d x}{c}\right )-2 p r t^2 u^2 \text {Li}_4\left (-\frac {b x}{a}\right )-2 q r t^2 u^2 \text {Li}_4\left (-\frac {d x}{c}\right )\\ \end {align*}
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Mathematica [B] time = 0.88, size = 839, normalized size = 3.20 \[ p r t^2 u^2 \log (a+b x) \log ^3(h x)-\frac {1}{3} p r t^2 u^2 \log \left (\frac {b x}{a}+1\right ) \log ^3(h x)+q r t^2 u^2 \log (c+d x) \log ^3(h x)-\frac {2}{3} t^2 u^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^3(h x)-\frac {1}{3} q r t^2 u^2 \log \left (\frac {d x}{c}+1\right ) \log ^3(h x)-p r t^2 u^2 \log (x) \log (a+b x) \log ^2(h x)-2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log ^2(h x)+p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {b x}{a}+1\right ) \log ^2(h x)-q r t^2 u^2 \log (x) \log (c+d x) \log ^2(h x)-2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log ^2(h x)+t^2 u^2 \log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(h x)+t u \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 ) \log ^2(h x)+q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {d x}{c}+1\right ) \log ^2(h x)+p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log (h x)+2 p r t u \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log (h x)-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {b x}{a}+1\right ) \log (h x)+q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log (h x)+2 q r t u \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log (h x)-2 t u \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 ) \log (h x)-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {d x}{c}+1\right ) \log (h x)-p r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-q r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+\log (x) \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 )-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {d x}{c}\right )-2 p r t^2 u^2 \text {Li}_4\left (-\frac {b x}{a}\right )-2 q r t^2 u^2 \text {Li}_4\left (-\frac {d x}{c}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) \ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,{\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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