Integrand size = 40, antiderivative size = 45 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \]
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 (b B c n-a B d n) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \]
Time = 0.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2973, 2961, 2739, 15}
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.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {c+d x}{(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}d\frac {a+b x}{c+d x}}{b c-a d}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3}{(a+b x)^3}d\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n (b c-a d)}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {(c+d x)^2}{2 B n (a+b x)^2 (b c-a d)}\) |
3.3.33.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
Time = 163.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
| default | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
| parallelrisch | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
| risch | \(\frac {2}{B n \left (a d -c b \right ) {\left (2 A +2 B \ln \left (e \right )+2 B \ln \left (\left (b x +a \right )^{n}\right )-2 B \ln \left (\left (d x +c \right )^{n}\right )-i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}-i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )+i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}\right )}^{2}}\) | \(366\) |
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (43) = 86\).
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 5.29 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (d x + c\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n \log \left (e\right )^{2} + 2 \, {\left (A B^{2} b c - A B^{2} a d\right )} n \log \left (e\right ) + {\left (A^{2} B b c - A^{2} B a d\right )} n + 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right ) + {\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (d x + c\right )\right )}} \]
-1/2/((B^3*b*c - B^3*a*d)*n^3*log(b*x + a)^2 + (B^3*b*c - B^3*a*d)*n^3*log (d*x + c)^2 + (B^3*b*c - B^3*a*d)*n*log(e)^2 + 2*(A*B^2*b*c - A*B^2*a*d)*n *log(e) + (A^2*B*b*c - A^2*B*a*d)*n + 2*((B^3*b*c - B^3*a*d)*n^2*log(e) + (A*B^2*b*c - A*B^2*a*d)*n^2)*log(b*x + a) - 2*((B^3*b*c - B^3*a*d)*n^3*log (b*x + a) + (B^3*b*c - B^3*a*d)*n^2*log(e) + (A*B^2*b*c - A*B^2*a*d)*n^2)* log(d*x + c))
Timed out. \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (43) = 86\).
Time = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 4.89 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} B^{3} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} A^{2} B + 2 \, {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} A B^{2} + {\left (b c n \log \left (e\right )^{2} - a d n \log \left (e\right )^{2}\right )} B^{3} + 2 \, {\left ({\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )\right )}} \]
-1/2/((b*c*n - a*d*n)*B^3*log((b*x + a)^n)^2 + (b*c*n - a*d*n)*B^3*log((d* x + c)^n)^2 + (b*c*n - a*d*n)*A^2*B + 2*(b*c*n*log(e) - a*d*n*log(e))*A*B^ 2 + (b*c*n*log(e)^2 - a*d*n*log(e)^2)*B^3 + 2*((b*c*n - a*d*n)*A*B^2 + (b* c*n*log(e) - a*d*n*log(e))*B^3)*log((b*x + a)^n) - 2*((b*c*n - a*d*n)*B^3* log((b*x + a)^n) + (b*c*n - a*d*n)*A*B^2 + (b*c*n*log(e) - a*d*n*log(e))*B ^3)*log((d*x + c)^n))
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (43) = 86\).
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 7.13 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left (B^{3} b c n^{3} \log \left (b x + a\right )^{2} - B^{3} a d n^{3} \log \left (b x + a\right )^{2} - 2 \, B^{3} b c n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + 2 \, B^{3} a d n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + B^{3} b c n^{3} \log \left (d x + c\right )^{2} - B^{3} a d n^{3} \log \left (d x + c\right )^{2} + 2 \, B^{3} b c n^{2} \log \left (b x + a\right ) \log \left (e\right ) - 2 \, B^{3} a d n^{2} \log \left (b x + a\right ) \log \left (e\right ) - 2 \, B^{3} b c n^{2} \log \left (d x + c\right ) \log \left (e\right ) + 2 \, B^{3} a d n^{2} \log \left (d x + c\right ) \log \left (e\right ) + 2 \, A B^{2} b c n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} a d n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} b c n^{2} \log \left (d x + c\right ) + 2 \, A B^{2} a d n^{2} \log \left (d x + c\right ) + B^{3} b c n \log \left (e\right )^{2} - B^{3} a d n \log \left (e\right )^{2} + 2 \, A B^{2} b c n \log \left (e\right ) - 2 \, A B^{2} a d n \log \left (e\right ) + A^{2} B b c n - A^{2} B a d n\right )}} \]
-1/2/(B^3*b*c*n^3*log(b*x + a)^2 - B^3*a*d*n^3*log(b*x + a)^2 - 2*B^3*b*c* n^3*log(b*x + a)*log(d*x + c) + 2*B^3*a*d*n^3*log(b*x + a)*log(d*x + c) + B^3*b*c*n^3*log(d*x + c)^2 - B^3*a*d*n^3*log(d*x + c)^2 + 2*B^3*b*c*n^2*lo g(b*x + a)*log(e) - 2*B^3*a*d*n^2*log(b*x + a)*log(e) - 2*B^3*b*c*n^2*log( d*x + c)*log(e) + 2*B^3*a*d*n^2*log(d*x + c)*log(e) + 2*A*B^2*b*c*n^2*log( b*x + a) - 2*A*B^2*a*d*n^2*log(b*x + a) - 2*A*B^2*b*c*n^2*log(d*x + c) + 2 *A*B^2*a*d*n^2*log(d*x + c) + B^3*b*c*n*log(e)^2 - B^3*a*d*n*log(e)^2 + 2* A*B^2*b*c*n*log(e) - 2*A*B^2*a*d*n*log(e) + A^2*B*b*c*n - A^2*B*a*d*n)
Time = 1.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=\frac {1}{2\,B\,n\,\left (a\,d-b\,c\right )\,\left (A^2+2\,A\,B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )+B^2\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\right )} \]