Integrand size = 47, antiderivative size = 150 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {x^2}{2}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d^2} \]
1/2*x^2-1/2*x*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/d-1/2*x*ln(1+b*exp(d* x+c)/(a+(a^2+b^2)^(1/2)))/d-1/2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) ))/d^2-1/2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/d^2
Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(150)=300\).
Time = 0.45 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.65 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {-a d x \log \left (1+\frac {\left (a-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-\sqrt {a^2+b^2} d x \log \left (1+\frac {\left (a-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a d x \log \left (1+\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-\sqrt {a^2+b^2} d x \log \left (1+\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a d x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-a d x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+\left (a+\sqrt {a^2+b^2}\right ) \operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+\left (-a+\sqrt {a^2+b^2}\right ) \operatorname {PolyLog}\left (2,-\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-a \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \sqrt {a^2+b^2} d^2} \]
(-(a*d*x*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b]) - Sqrt[a^2 + b^2 ]*d*x*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] + a*d*x*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - Sqrt[a^2 + b^2]*d*x*Log[1 + ((a + Sq rt[a^2 + b^2])*E^(-c - d*x))/b] + a*d*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[ a^2 + b^2])] - a*d*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + (a + Sqrt[a^2 + b^2])*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] + (- a + Sqrt[a^2 + b^2])*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] + a*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - a*PolyLog[2, -(( b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*Sqrt[a^2 + b^2]*d^2)
Time = 1.12 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2695, 27, 2615, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {x \left (b e-a e e^{c+d x}\right )}{-2 a e e^{c+d x}-b e e^{2 (c+d x)}+b e} \, dx\) |
\(\Big \downarrow \) 2695 |
\(\displaystyle -\left (e \left (a-\sqrt {a^2+b^2}\right ) \int -\frac {x}{2 \left (b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e\right )}dx\right )-e \left (\sqrt {a^2+b^2}+a\right ) \int -\frac {x}{2 \left (b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \int \frac {x}{b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e}dx+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \int \frac {x}{b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e}dx\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \int \frac {e^{c+d x} x}{b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e}dx}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \int \frac {e^{c+d x} x}{b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e}dx}{\sqrt {a^2+b^2}+a}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}-\frac {\int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}-\frac {\int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d e}\right )}{\sqrt {a^2+b^2}+a}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2 e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2 e}\right )}{\sqrt {a^2+b^2}+a}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 e}+\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2 e}+\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}\right )}{\sqrt {a^2+b^2}+a}\right )\) |
((a - Sqrt[a^2 + b^2])*e*(x^2/(2*(a - Sqrt[a^2 + b^2])*e) - (b*((x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d*e) + PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]/(b*d^2*e)))/(a - Sqrt[a^2 + b^2])))/2 + ((a + Sqrt[a^2 + b^2])*e*(x^2/(2*(a + Sqrt[a^2 + b^2])*e) - (b*((x*Log[1 + (b* E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d*e) + PolyLog[2, -((b*E^(c + d*x) )/(a + Sqrt[a^2 + b^2]))]/(b*d^2*e)))/(a + Sqrt[a^2 + b^2])))/2
3.6.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F _)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(Simplify[(2*c*h - b*i)/q] + i) Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Simp[(Simplify[(2*c*h - b*i)/q] - i) Int[(f + g*x)^m/(b + q + 2*c*F^ u), x], x]] /; FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ [u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(128)=256\).
Time = 0.11 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.12
method | result | size |
risch | \(\frac {c \ln \left (2 \,{\mathrm e}^{d x +c} a +{\mathrm e}^{2 d x +2 c} b -b \right )}{2 d^{2}}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2}}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{2 d}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{2 d^{2}}-\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{2 d}-\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{2 d^{2}}-\frac {\operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{2 d^{2}}-\frac {\operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{2 d^{2}}+\frac {x^{2}}{2}+\frac {c x}{d}+\frac {c^{2}}{2 d^{2}}\) | \(318\) |
default | \(\text {Expression too large to display}\) | \(965\) |
int((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x,met hod=_RETURNVERBOSE)
1/2/d^2*c*ln(2*exp(d*x+c)*a+exp(2*d*x+2*c)*b-b)-1/d^2*c*ln(exp(d*x+c))-1/2 /d*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/2/d^2*ln ((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/2/d*ln((b*exp (d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/2/d^2*ln((b*exp(d*x+c) +(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/2/d^2*dilog((-b*exp(d*x+c)+(a ^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/2/d^2*dilog((b*exp(d*x+c)+(a^2+b^ 2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/2*x^2+1/d*c*x+1/2/d^2*c^2
Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.67 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {d^{2} x^{2} + c \log \left (2 \, b e^{\left (d x + c\right )} + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + c \log \left (2 \, b e^{\left (d x + c\right )} - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - {\left (d x + c\right )} \log \left (-\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} + a e^{\left (d x + c\right )} - b}{b}\right ) - {\left (d x + c\right )} \log \left (\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} - a e^{\left (d x + c\right )} + b}{b}\right ) - {\rm Li}_2\left (\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} + a e^{\left (d x + c\right )} - b}{b} + 1\right ) - {\rm Li}_2\left (-\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} - a e^{\left (d x + c\right )} + b}{b} + 1\right )}{2 \, d^{2}} \]
integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) ,x, algorithm="fricas")
1/2*(d^2*x^2 + c*log(2*b*e^(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + c*log(2*b*e^(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (d*x + c)*log(- (b*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) + a*e^(d*x + c) - b)/b) - (d*x + c)*l og((b*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) - a*e^(d*x + c) + b)/b) - dilog((b *sqrt((a^2 + b^2)/b^2)*e^(d*x + c) + a*e^(d*x + c) - b)/b + 1) - dilog(-(b *sqrt((a^2 + b^2)/b^2)*e^(d*x + c) - a*e^(d*x + c) + b)/b + 1))/d^2
\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int \frac {x \left (a e^{c} e^{d x} - b\right )}{2 a e^{c} e^{d x} + b e^{2 c} e^{2 d x} - b}\, dx \]
\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int { \frac {{\left (a e e^{\left (d x + c\right )} - b e\right )} x}{b e e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e e^{\left (d x + c\right )} - b e} \,d x } \]
integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) ,x, algorithm="maxima")
\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int { \frac {{\left (a e e^{\left (d x + c\right )} - b e\right )} x}{b e e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e e^{\left (d x + c\right )} - b e} \,d x } \]
integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) ,x, algorithm="giac")
Timed out. \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int -\frac {x\,\left (b\,e-a\,e\,{\mathrm {e}}^{c+d\,x}\right )}{2\,a\,e\,{\mathrm {e}}^{c+d\,x}-b\,e+b\,e\,{\mathrm {e}}^{2\,c+2\,d\,x}} \,d x \]