3.1.57 \(\int \frac {\log ^2(i (j (h x)^t)^u) \log (e (f (a+b x)^p (c+d x)^q)^r)}{x} \, dx\) [57]

Optimal. Leaf size=262 \[ -\frac {p r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {d x}{c}\right )-2 p r t^2 u^2 \text {Li}_4\left (-\frac {b x}{a}\right )-2 q r t^2 u^2 \text {Li}_4\left (-\frac {d x}{c}\right ) \]

[Out]

-1/3*p*r*ln(i*(j*(h*x)^t)^u)^3*ln(1+b*x/a)/t/u+1/3*ln(i*(j*(h*x)^t)^u)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/t/u-1
/3*q*r*ln(i*(j*(h*x)^t)^u)^3*ln(1+d*x/c)/t/u-p*r*ln(i*(j*(h*x)^t)^u)^2*polylog(2,-b*x/a)-q*r*ln(i*(j*(h*x)^t)^
u)^2*polylog(2,-d*x/c)+2*p*r*t*u*ln(i*(j*(h*x)^t)^u)*polylog(3,-b*x/a)+2*q*r*t*u*ln(i*(j*(h*x)^t)^u)*polylog(3
,-d*x/c)-2*p*r*t^2*u^2*polylog(4,-b*x/a)-2*q*r*t^2*u^2*polylog(4,-d*x/c)

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Rubi [A]
time = 0.65, 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 = {2585, 2354, 2421, 2430, 6724, 2495} \begin {gather*} -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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/x,x]

[Out]

-1/3*(p*r*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (b*x)/a])/(t*u) + (Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d
*x)^q)^r])/(3*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (d*x)/c])/(3*t*u) - p*r*Log[i*(j*(h*x)^t)^u]^2*PolyLo
g[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((d*x)/c)] + 2*p*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[
3, -((b*x)/a)] + 2*q*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[3, -((d*x)/c)] - 2*p*r*t^2*u^2*PolyLog[4, -((b*x)/a)]
- 2*q*r*t^2*u^2*PolyLog[4, -((d*x)/c)]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[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]

Rule 2421

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] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[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]

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2495

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]]

Rule 2585

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] + (-Dist[b*p*(r/(k*n*t*(m + 1))), Int[(s + t*Log[i*
(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

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 &=\text {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 )\\ &=\text {Subst}\left (\text {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}-\text {Subst}\left (\text {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 )-\text {Subst}\left (\text {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}+\text {Subst}\left (\text {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 )+\text {Subst}\left (\text {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 )+\text {Subst}\left (\text {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 )+\text {Subst}\left (\text {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 )-\text {Subst}\left (\text {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 )-\text {Subst}\left (\text {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] Leaf count is larger than twice the leaf count of optimal. \(839\) vs. \(2(262)=524\).
time = 0.60, size = 839, normalized size = 3.20 \begin {gather*} -p r t^2 u^2 \log (x) \log ^2(h x) \log (a+b x)+p r t^2 u^2 \log ^3(h x) \log (a+b x)+2 p r t u \log (x) \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-2 p r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-p r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+p r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-\frac {1}{3} p r t^2 u^2 \log ^3(h x) \log \left (1+\frac {b x}{a}\right )+p r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )-p r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )-q r t^2 u^2 \log (x) \log ^2(h x) \log (c+d x)+q r t^2 u^2 \log ^3(h x) \log (c+d x)+2 q r t u \log (x) \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-2 q r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-q r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+q r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+t^2 u^2 \log (x) \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {2}{3} t^2 u^2 \log ^3(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 t u \log (x) \log (h 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 )+t u \log ^2(h 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 (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 )-\frac {1}{3} q r t^2 u^2 \log ^3(h x) \log \left (1+\frac {d x}{c}\right )+q r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )-q r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/x,x]

[Out]

-(p*r*t^2*u^2*Log[x]*Log[h*x]^2*Log[a + b*x]) + p*r*t^2*u^2*Log[h*x]^3*Log[a + b*x] + 2*p*r*t*u*Log[x]*Log[h*x
]*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] - 2*p*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] - p*r*Log[x]*Log[
i*(j*(h*x)^t)^u]^2*Log[a + b*x] + p*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^2*Log[a + b*x] - (p*r*t^2*u^2*Log[h*x]^3*L
og[1 + (b*x)/a])/3 + p*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[1 + (b*x)/a] - p*r*Log[h*x]*Log[i*(j*(h*x)^t)
^u]^2*Log[1 + (b*x)/a] - q*r*t^2*u^2*Log[x]*Log[h*x]^2*Log[c + d*x] + q*r*t^2*u^2*Log[h*x]^3*Log[c + d*x] + 2*
q*r*t*u*Log[x]*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[c + d*x] - 2*q*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[c +
d*x] - q*r*Log[x]*Log[i*(j*(h*x)^t)^u]^2*Log[c + d*x] + q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^2*Log[c + d*x] + t^2
*u^2*Log[x]*Log[h*x]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - (2*t^2*u^2*Log[h*x]^3*Log[e*(f*(a + b*x)^p*(c +
d*x)^q)^r])/3 - 2*t*u*Log[x]*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + t*u*Log[h*x]
^2*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + Log[x]*Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x
)^p*(c + d*x)^q)^r] - (q*r*t^2*u^2*Log[h*x]^3*Log[1 + (d*x)/c])/3 + q*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Lo
g[1 + (d*x)/c] - q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c] - p*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2,
-((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((d*x)/c)] + 2*p*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[3, -(
(b*x)/a)] + 2*q*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[3, -((d*x)/c)] - 2*p*r*t^2*u^2*PolyLog[4, -((b*x)/a)] - 2*q
*r*t^2*u^2*PolyLog[4, -((d*x)/c)]

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Maple [F]
time = 1.07, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(i*(j*(h*x)^t)^u)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)

[Out]

int(ln(i*(j*(h*x)^t)^u)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="maxima")

[Out]

1/12*(4*t^2*u^2*log(x)^3 - 6*(2*t^2*u^2*log(h) + 2*t*u^2*log(j) + I*pi*t*u)*log(x)^2 + 12*log(x)*log((x^t)^u)^
2 + 3*(4*t^2*u^2*log(h)^2 + 8*t*u^2*log(h)*log(j) + 4*u^2*log(j)^2 - pi^2 + 4*I*pi*(t*u*log(h) + u*log(j)))*lo
g(x) - 12*(t*u*log(x)^2 + (-I*pi - 2*t*u*log(h) - 2*u*log(j))*log(x))*log((x^t)^u))*log(((b*x + a)^p)^r) + 1/1
2*(4*t^2*u^2*log(x)^3 - 6*(2*t^2*u^2*log(h) + 2*t*u^2*log(j) + I*pi*t*u)*log(x)^2 + 12*log(x)*log((x^t)^u)^2 +
3*(4*t^2*u^2*log(h)^2 + 8*t*u^2*log(h)*log(j) + 4*u^2*log(j)^2 - pi^2 + 4*I*pi*(t*u*log(h) + u*log(j)))*log(x
) - 12*(t*u*log(x)^2 + (-I*pi - 2*t*u*log(h) - 2*u*log(j))*log(x))*log((x^t)^u))*log(((d*x + c)^q)^r) - integr
ate(1/12*(3*pi^2*(r*log(f) + 1)*a*c - 12*I*pi*(t*u*log(h) + (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))*a*c
+ 4*((p*r*t^2*u^2 + q*r*t^2*u^2)*b*d*x^2 + (b*c*p*r*t^2*u^2 + a*d*q*r*t^2*u^2)*x)*log(x)^3 - 12*(t^2*u^2*log(
h)^2 + 2*t*u^2*log(h)*log(j) + u^2*log(j)^2 + (r*t^2*u^2*log(h)^2 + 2*r*t*u^2*log(h)*log(j) + r*u^2*log(j)^2)*
log(f))*a*c + 3*(pi^2*(r*log(f) + 1)*b*d - 4*I*pi*(t*u*log(h) + (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))
*b*d - 4*(t^2*u^2*log(h)^2 + 2*t*u^2*log(h)*log(j) + u^2*log(j)^2 + (r*t^2*u^2*log(h)^2 + 2*r*t*u^2*log(h)*log
(j) + r*u^2*log(j)^2)*log(f))*b*d)*x^2 - 12*((r*log(f) + 1)*b*d*x^2 + (r*log(f) + 1)*a*c + ((r*log(f) + 1)*b*c
+ (r*log(f) + 1)*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 - 6*((I*pi*(p*
r*t*u + q*r*t*u)*b*d + 2*((p*r*t^2*u^2 + q*r*t^2*u^2)*log(h) + (p*r*t*u^2 + q*r*t*u^2)*log(j))*b*d)*x^2 + (2*(
p*r*t^2*u^2*log(h) + p*r*t*u^2*log(j))*b*c + 2*(q*r*t^2*u^2*log(h) + q*r*t*u^2*log(j))*a*d + I*pi*(b*c*p*r*t*u
+ a*d*q*r*t*u))*x)*log(x)^2 + 3*(pi^2*((r*log(f) + 1)*b*c + (r*log(f) + 1)*a*d) - 4*(t^2*u^2*log(h)^2 + 2*t*u
^2*log(h)*log(j) + u^2*log(j)^2 + (r*t^2*u^2*log(h)^2 + 2*r*t*u^2*log(h)*log(j) + r*u^2*log(j)^2)*log(f))*b*c
- 4*(t^2*u^2*log(h)^2 + 2*t*u^2*log(h)*log(j) + u^2*log(j)^2 + (r*t^2*u^2*log(h)^2 + 2*r*t*u^2*log(h)*log(j) +
r*u^2*log(j)^2)*log(f))*a*d - 4*I*pi*((t*u*log(h) + (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))*b*c + (t*u
*log(h) + (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))*a*d))*x - 12*(I*pi*(r*log(f) + 1)*a*c + 2*(t*u*log(h)
+ (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))*a*c + (I*pi*(r*log(f) + 1)*b*d + 2*(t*u*log(h) + (r*t*u*log(
h) + r*u*log(j))*log(f) + u*log(j))*b*d)*x^2 + ((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)*l
og(x)^2 + (2*(t*u*log(h) + (r*t*u*log(h) + r*u*log(j))*log(f) + u*log(j))*b*c + 2*(t*u*log(h) + (r*t*u*log(h)
+ r*u*log(j))*log(f) + u*log(j))*a*d + I*pi*((r*log(f) + 1)*b*c + (r*log(f) + 1)*a*d))*x + ((-I*pi*(p*r + q*r)
*b*d - 2*((p*r*t*u + q*r*t*u)*log(h) + (p*r*u + q*r*u)*log(j))*b*d)*x^2 - (2*(p*r*t*u*log(h) + p*r*u*log(j))*b
*c + 2*(q*r*t*u*log(h) + q*r*u*log(j))*a*d + I*pi*(b*c*p*r + a*d*q*r))*x)*log(x))*log((x^t)^u) - 3*((pi^2*(p*r
+ q*r)*b*d - 4*I*pi*((p*r*t*u + q*r*t*u)*log(h) + (p*r*u + q*r*u)*log(j))*b*d - 4*((p*r*t^2*u^2 + q*r*t^2*u^2
)*log(h)^2 + 2*(p*r*t*u^2 + q*r*t*u^2)*log(h)*log(j) + (p*r*u^2 + q*r*u^2)*log(j)^2)*b*d)*x^2 + (pi^2*(b*c*p*r
+ a*d*q*r) - 4*(p*r*t^2*u^2*log(h)^2 + 2*p*r*t*u^2*log(h)*log(j) + p*r*u^2*log(j)^2)*b*c - 4*(q*r*t^2*u^2*log
(h)^2 + 2*q*r*t*u^2*log(h)*log(j) + q*r*u^2*log(j)^2)*a*d - 4*I*pi*((p*r*t*u*log(h) + p*r*u*log(j))*b*c + (q*r
*t*u*log(h) + q*r*u*log(j))*a*d))*x)*log(x))/(b*d*x^3 + a*c*x + (b*c + a*d)*x^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="fricas")

[Out]

integral(-1/4*(pi^2*r*log(f) - 4*(p*r*t^2*u^2*log(b*x + a) + q*r*t^2*u^2*log(d*x + c) + r*t^2*u^2*log(f) + t^2
*u^2)*log(h*x)^2 - 4*(r*u^2*log(f) + u^2)*log(j)^2 + pi^2 - (4*p*r*u^2*log(j)^2 + 4*I*pi*p*r*u*log(j) - pi^2*p
*r)*log(b*x + a) - (4*q*r*u^2*log(j)^2 + 4*I*pi*q*r*u*log(j) - pi^2*q*r)*log(d*x + c) - 4*(I*pi*r*t*u*log(f) +
I*pi*t*u + (2*p*r*t*u^2*log(j) + I*pi*p*r*t*u)*log(b*x + a) + (2*q*r*t*u^2*log(j) + I*pi*q*r*t*u)*log(d*x + c
) + 2*(r*t*u^2*log(f) + t*u^2)*log(j))*log(h*x) - 4*(I*pi*r*u*log(f) + I*pi*u)*log(j))/x, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(i*(j*(h*x)**t)**u)**2*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/x,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6438 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(I*((h*x)^t*j)^u)^2/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^2)/x,x)

[Out]

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^2)/x, x)

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