\(\int \frac {15+e^{2 x} (15+5 x^2)+e^x (-30-5 x^2-5 x^3)}{(3 x+x^2 \log (3)+e^x (-6 x+x^3-2 x^2 \log (3))+e^{2 x} (3 x-x^3+x^2 \log (3))) \log ^2(\frac {-3-x \log (3)+e^x (3-x^2+x \log (3))}{-x+e^x x})} \, dx\) [3053]

Optimal result

Integrand size = 120, antiderivative size = 27 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (\frac {3}{x}-x+\frac {x}{1-e^x}+\log (3)\right )} \]

[Out]

5/ln(ln(3)+x/(1-exp(x))+3/x-x)

Rubi [F]

\[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx \]

[In]

Int[(15 + E^(2*x)*(15 + 5*x^2) + E^x*(-30 - 5*x^2 - 5*x^3))/((3*x + x^2*Log[3] + E^x*(-6*x + x^3 - 2*x^2*Log[3
]) + E^(2*x)*(3*x - x^3 + x^2*Log[3]))*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/(-x + E^x*x)]^2),x]

[Out]

5*Defer[Int][1/((-1 + E^x)*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x] + 5*Defer[Int
][1/(x*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x] + 5*(3 + Log[3])*Defer[Int][1/((3
- 3*E^x + E^x*x^2 + x*Log[3] - E^x*x*Log[3])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2
), x] + 5*Log[3]*Defer[Int][x/((3 - 3*E^x + E^x*x^2 + x*Log[3] - E^x*x*Log[3])*Log[(-3 - x*Log[3] + E^x*(3 - x
^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x] - 10*(1 + Log[3]/Sqrt[12 + Log[3]^2])*Defer[Int][1/((2*x - Log[3] - Sqr
t[12 + Log[3]^2])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x] + 5*(6 + Log[3]^2)*(1
+ Log[3]/Sqrt[12 + Log[3]^2])*Defer[Int][1/((3 - 3*E^x + E^x*x^2 + x*Log[3] - E^x*x*Log[3])*(2*x - Log[3] - Sq
rt[12 + Log[3]^2])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x] - 10*(1 - Log[3]/Sqrt
[12 + Log[3]^2])*Defer[Int][1/((2*x - Log[3] + Sqrt[12 + Log[3]^2])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[
3]))/((-1 + E^x)*x)]^2), x] + 5*(6 + Log[3]^2)*(1 - Log[3]/Sqrt[12 + Log[3]^2])*Defer[Int][1/((3 - 3*E^x + E^x
*x^2 + x*Log[3] - E^x*x*Log[3])*(2*x - Log[3] + Sqrt[12 + Log[3]^2])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log
[3]))/((-1 + E^x)*x)]^2), x] - (10*Log[3]*Defer[Int][1/((-2*x + Log[3] + Sqrt[12 + Log[3]^2])*Log[(-3 - x*Log[
3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x])/Sqrt[12 + Log[3]^2] - (10*Log[27]*Defer[Int][1/((3 - 3*
E^x + E^x*x^2 + x*Log[3] - E^x*x*Log[3])*(-2*x + Log[3] + Sqrt[12 + Log[3]^2])*Log[(-3 - x*Log[3] + E^x*(3 - x
^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x])/Sqrt[12 + Log[3]^2] - (20*Log[3]*Defer[Int][1/((4*x + 2*Sqrt[12 + Log[
3]^2] - Log[9])*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x])/Sqrt[12 + Log[3]^2] - (
20*Log[27]*Defer[Int][1/((3 - 3*E^x + E^x*x^2 + x*Log[3] - E^x*x*Log[3])*(4*x + 2*Sqrt[12 + Log[3]^2] - Log[9]
)*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]^2), x])/Sqrt[12 + Log[3]^2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (1-e^x\right ) x \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx \\ & = \int \left (\frac {5}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}-\frac {5 \left (3+x^2\right )}{x \left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {5 \left (9-6 x (1-\log (3))-x^3 \log (3)-x^2 \left (3+\log (3)-\log ^2(3)\right )\right )}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-5 \int \frac {3+x^2}{x \left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {9-6 x (1-\log (3))-x^3 \log (3)-x^2 \left (3+\log (3)-\log ^2(3)\right )}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = -\left (5 \int \left (-\frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {2 x-\log (3)}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx\right )+5 \int \left (\frac {3 \left (1+\frac {\log (3)}{3}\right )}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {x \log (3)}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {-x \left (6+\log ^2(3)\right )-\log (27)}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-5 \int \frac {2 x-\log (3)}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {-x \left (6+\log ^2(3)\right )-\log (27)}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = -\left (5 \int \left (\frac {2 x}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {\log (3)}{\left (3-x^2+x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx\right )+5 \int \left (\frac {x \left (6+\log ^2(3)\right )}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {\log (27)}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-10 \int \frac {x}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-(5 \log (3)) \int \frac {1}{\left (3-x^2+x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right )\right ) \int \frac {x}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (27)) \int \frac {1}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-10 \int \left (\frac {1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx-(5 \log (3)) \int \left (\frac {2}{\sqrt {12+\log ^2(3)} \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {4}{\sqrt {12+\log ^2(3)} \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right )\right ) \int \left (\frac {1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+(5 \log (27)) \int \left (-\frac {2}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \sqrt {12+\log ^2(3)} \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}-\frac {4}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \sqrt {12+\log ^2(3)} \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\frac {(10 \log (3)) \int \frac {1}{\left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\frac {(20 \log (3)) \int \frac {1}{\left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\left (10 \left (1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right ) \left (1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\left (10 \left (1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right ) \left (1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\frac {(10 \log (27)) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\frac {(20 \log (27)) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \]

[In]

Integrate[(15 + E^(2*x)*(15 + 5*x^2) + E^x*(-30 - 5*x^2 - 5*x^3))/((3*x + x^2*Log[3] + E^x*(-6*x + x^3 - 2*x^2
*Log[3]) + E^(2*x)*(3*x - x^3 + x^2*Log[3]))*Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/(-x + E^x*x)]^2),x
]

[Out]

5/Log[(-3 - x*Log[3] + E^x*(3 - x^2 + x*Log[3]))/((-1 + E^x)*x)]

Maple [A] (verified)

Time = 9.75 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37

method	result	size
parallelrisch	\(\frac {5}{\ln \left (\frac {\left (x \ln \left (3\right )-x^{2}+3\right ) {\mathrm e}^{x}-x \ln \left (3\right )-3}{\left ({\mathrm e}^{x}-1\right ) x}\right )}\)	\(37\)
risch	\(\frac {10 i}{\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right ) \operatorname {csgn}\left (i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )\right ) \operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{2}+\pi \,\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right ) \operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{3}-\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \ln \left (x \right )-2 i \ln \left ({\mathrm e}^{x}-1\right )+2 i \ln \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}\)	\(483\)

[In]

int(((5*x^2+15)*exp(x)^2+(-5*x^3-5*x^2-30)*exp(x)+15)/((x^2*ln(3)-x^3+3*x)*exp(x)^2+(-2*x^2*ln(3)+x^3-6*x)*exp
(x)+x^2*ln(3)+3*x)/ln(((x*ln(3)-x^2+3)*exp(x)-x*ln(3)-3)/(exp(x)*x-x))^2,x,method=_RETURNVERBOSE)

[Out]

5/ln(((x*ln(3)-x^2+3)*exp(x)-x*ln(3)-3)/(exp(x)-1)/x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-\frac {{\left (x^{2} - x \log \left (3\right ) - 3\right )} e^{x} + x \log \left (3\right ) + 3}{x e^{x} - x}\right )} \]

[In]

integrate(((5*x^2+15)*exp(x)^2+(-5*x^3-5*x^2-30)*exp(x)+15)/((x^2*log(3)-x^3+3*x)*exp(x)^2+(-2*x^2*log(3)+x^3-
6*x)*exp(x)+x^2*log(3)+3*x)/log(((x*log(3)-x^2+3)*exp(x)-x*log(3)-3)/(exp(x)*x-x))^2,x, algorithm="fricas")

[Out]

5/log(-((x^2 - x*log(3) - 3)*e^x + x*log(3) + 3)/(x*e^x - x))

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log {\left (\frac {- x \log {\left (3 \right )} + \left (- x^{2} + x \log {\left (3 \right )} + 3\right ) e^{x} - 3}{x e^{x} - x} \right )}} \]

[In]

integrate(((5*x**2+15)*exp(x)**2+(-5*x**3-5*x**2-30)*exp(x)+15)/((x**2*ln(3)-x**3+3*x)*exp(x)**2+(-2*x**2*ln(3
)+x**3-6*x)*exp(x)+x**2*ln(3)+3*x)/ln(((x*ln(3)-x**2+3)*exp(x)-x*ln(3)-3)/(exp(x)*x-x))**2,x)

[Out]

5/log((-x*log(3) + (-x**2 + x*log(3) + 3)*exp(x) - 3)/(x*exp(x) - x))

Maxima [A] (verification not implemented)

none

Time = 1.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-{\left (x^{2} - x \log \left (3\right ) - 3\right )} e^{x} - x \log \left (3\right ) - 3\right ) - \log \left (x\right ) - \log \left (e^{x} - 1\right )} \]

[In]

integrate(((5*x^2+15)*exp(x)^2+(-5*x^3-5*x^2-30)*exp(x)+15)/((x^2*log(3)-x^3+3*x)*exp(x)^2+(-2*x^2*log(3)+x^3-
6*x)*exp(x)+x^2*log(3)+3*x)/log(((x*log(3)-x^2+3)*exp(x)-x*log(3)-3)/(exp(x)*x-x))^2,x, algorithm="maxima")

[Out]

5/(log(-(x^2 - x*log(3) - 3)*e^x - x*log(3) - 3) - log(x) - log(e^x - 1))

Giac [A] (verification not implemented)

none

Time = 0.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-x^{2} e^{x} + x e^{x} \log \left (3\right ) - x \log \left (3\right ) + 3 \, e^{x} - 3\right ) - \log \left (x e^{x} - x\right )} \]

[In]

integrate(((5*x^2+15)*exp(x)^2+(-5*x^3-5*x^2-30)*exp(x)+15)/((x^2*log(3)-x^3+3*x)*exp(x)^2+(-2*x^2*log(3)+x^3-
6*x)*exp(x)+x^2*log(3)+3*x)/log(((x*log(3)-x^2+3)*exp(x)-x*log(3)-3)/(exp(x)*x-x))^2,x, algorithm="giac")

[Out]

5/(log(-x^2*e^x + x*e^x*log(3) - x*log(3) + 3*e^x - 3) - log(x*e^x - x))

Mupad [B] (verification not implemented)

Time = 10.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\ln \left (\frac {x\,\ln \left (3\right )-{\mathrm {e}}^x\,\left (-x^2+\ln \left (3\right )\,x+3\right )+3}{x-x\,{\mathrm {e}}^x}\right )} \]

[In]

int((exp(2*x)*(5*x^2 + 15) - exp(x)*(5*x^2 + 5*x^3 + 30) + 15)/(log((x*log(3) - exp(x)*(x*log(3) - x^2 + 3) +
3)/(x - x*exp(x)))^2*(3*x - exp(x)*(6*x + 2*x^2*log(3) - x^3) + x^2*log(3) + exp(2*x)*(3*x + x^2*log(3) - x^3)
)),x)

[Out]

5/log((x*log(3) - exp(x)*(x*log(3) - x^2 + 3) + 3)/(x - x*exp(x)))