3.33.28 \(\int \frac {-2 x-e^5 x-x^2+e^x (-6 e^5 x^3-6 x^4)+e^{2 x} (-6 e^5 x^3-6 x^4)+(e^5+x+e^x (-18 e^5 x^2-18 x^3)+e^{2 x} (-18 e^5 x^2-18 x^3)) \log (e^5+x)+(e^x (-18 e^5 x-18 x^2)+e^{2 x} (-18 e^5 x-18 x^2)) \log ^2(e^5+x)+(e^x (-6 e^5-6 x)+e^{2 x} (-6 e^5-6 x)) \log ^3(e^5+x)}{e^5 x^3+x^4+(3 e^5 x^2+3 x^3) \log (e^5+x)+(3 e^5 x+3 x^2) \log ^2(e^5+x)+(e^5+x) \log ^3(e^5+x)} \, dx\)

Optimal. Leaf size=22 \[ -3 \left (1+e^x\right )^2+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \]

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Rubi [F]  time = 2.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 x-e^5 x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5*x^3 - 6*x^4) + (E^5 + x + E^x*(-18*E^5*x^
2 - 18*x^3) + E^(2*x)*(-18*E^5*x^2 - 18*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^5*x -
18*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x))*Log[E^5 + x]^3)/(E^5*x^3 + x^4 + (3*E^
5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E^5*x + 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]

[Out]

-6*E^x - 3*E^(2*x) + E^5*Defer[Int][(x + Log[E^5 + x])^(-3), x] - (2 + E^5)*Defer[Int][(x + Log[E^5 + x])^(-3)
, x] - 2*Defer[Int][x/(x + Log[E^5 + x])^3, x] - E^10*Defer[Int][1/((E^5 + x)*(x + Log[E^5 + x])^3), x] + E^5*
(2 + E^5)*Defer[Int][1/((E^5 + x)*(x + Log[E^5 + x])^3), x] + Defer[Int][(x + Log[E^5 + x])^(-2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-2-e^5\right ) x-x^2+e^x \left (-6 e^5 x^3-6 x^4\right )+e^{2 x} \left (-6 e^5 x^3-6 x^4\right )+\left (e^5+x+e^x \left (-18 e^5 x^2-18 x^3\right )+e^{2 x} \left (-18 e^5 x^2-18 x^3\right )\right ) \log \left (e^5+x\right )+\left (e^x \left (-18 e^5 x-18 x^2\right )+e^{2 x} \left (-18 e^5 x-18 x^2\right )\right ) \log ^2\left (e^5+x\right )+\left (e^x \left (-6 e^5-6 x\right )+e^{2 x} \left (-6 e^5-6 x\right )\right ) \log ^3\left (e^5+x\right )}{e^5 x^3+x^4+\left (3 e^5 x^2+3 x^3\right ) \log \left (e^5+x\right )+\left (3 e^5 x+3 x^2\right ) \log ^2\left (e^5+x\right )+\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx\\ &=\int \frac {-x \left (2+e^5+x+6 e^{5+x} x^2+6 e^{5+2 x} x^2+6 e^x x^3+6 e^{2 x} x^3\right )-\left (e^5+x\right ) \left (-1+18 e^x x^2+18 e^{2 x} x^2\right ) \log \left (e^5+x\right )-18 e^x \left (1+e^x\right ) x \left (e^5+x\right ) \log ^2\left (e^5+x\right )-6 e^x \left (1+e^x\right ) \left (e^5+x\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx\\ &=\int \left (-6 e^x-6 e^{2 x}-\frac {2 \left (1+\frac {e^5}{2}\right ) x}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}-\frac {x^2}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}+\frac {\log \left (e^5+x\right )}{\left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx\\ &=-\left (6 \int e^x \, dx\right )-6 \int e^{2 x} \, dx-\left (2+e^5\right ) \int \frac {x}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-\int \frac {x^2}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx+\int \frac {\log \left (e^5+x\right )}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx\\ &=-6 e^x-3 e^{2 x}-\left (2+e^5\right ) \int \left (\frac {1}{\left (x+\log \left (e^5+x\right )\right )^3}-\frac {e^5}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx-\int \left (-\frac {e^5}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {e^{10}}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3}\right ) \, dx+\int \left (-\frac {x}{\left (x+\log \left (e^5+x\right )\right )^3}+\frac {1}{\left (x+\log \left (e^5+x\right )\right )^2}\right ) \, dx\\ &=-6 e^x-3 e^{2 x}+e^5 \int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx-e^{10} \int \frac {1}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-\left (2+e^5\right ) \int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx+\left (e^5 \left (2+e^5\right )\right ) \int \frac {1}{\left (e^5+x\right ) \left (x+\log \left (e^5+x\right )\right )^3} \, dx-2 \int \frac {x}{\left (x+\log \left (e^5+x\right )\right )^3} \, dx+\int \frac {1}{\left (x+\log \left (e^5+x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 25, normalized size = 1.14 \begin {gather*} -6 e^x-3 e^{2 x}+\frac {x}{\left (x+\log \left (e^5+x\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5*x^3 - 6*x^4) + (E^5 + x + E^x*(-18*
E^5*x^2 - 18*x^3) + E^(2*x)*(-18*E^5*x^2 - 18*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^
5*x - 18*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x))*Log[E^5 + x]^3)/(E^5*x^3 + x^4 +
(3*E^5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E^5*x + 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]

[Out]

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

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fricas [B]  time = 0.71, size = 80, normalized size = 3.64 \begin {gather*} -\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="fricas")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 3*(e^(2*x) + 2*e^x)*log(x + e^5)^2 + 6*(x*e^(2*x) + 2*x*e^x)*log(x + e^5) - x)/(
x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

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giac [B]  time = 1.11, size = 89, normalized size = 4.05 \begin {gather*} -\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 6 \, x e^{\left (2 \, x\right )} \log \left (x + e^{5}\right ) + 12 \, x e^{x} \log \left (x + e^{5}\right ) + 3 \, e^{\left (2 \, x\right )} \log \left (x + e^{5}\right )^{2} + 6 \, e^{x} \log \left (x + e^{5}\right )^{2} - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="giac")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 6*x*e^(2*x)*log(x + e^5) + 12*x*e^x*log(x + e^5) + 3*e^(2*x)*log(x + e^5)^2 + 6*
e^x*log(x + e^5)^2 - x)/(x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

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maple [A]  time = 0.36, size = 23, normalized size = 1.05

method	result	size
risch	\(-3 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+\frac {x}{\left (\ln \left ({\mathrm e}^{5}+x \right )+x \right )^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*e
xp(5)-18*x^2)*exp(x))*ln(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+exp(5)+x
)*ln(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)+x)*ln(ex
p(5)+x)^3+(3*x*exp(5)+3*x^2)*ln(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*ln(exp(5)+x)+x^3*exp(5)+x^4),x,method=_RETURN
VERBOSE)

[Out]

-3*exp(2*x)-6*exp(x)+x/(ln(exp(5)+x)+x)^2

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maxima [B]  time = 0.81, size = 80, normalized size = 3.64 \begin {gather*} -\frac {3 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{x} + 3 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x}\right )} \log \left (x + e^{5}\right )^{2} + 6 \, {\left (x e^{\left (2 \, x\right )} + 2 \, x e^{x}\right )} \log \left (x + e^{5}\right ) - x}{x^{2} + 2 \, x \log \left (x + e^{5}\right ) + \log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="maxima")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 3*(e^(2*x) + 2*e^x)*log(x + e^5)^2 + 6*(x*e^(2*x) + 2*x*e^x)*log(x + e^5) - x)/(
x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

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mupad [B]  time = 2.70, size = 181, normalized size = 8.23 \begin {gather*} \frac {\frac {x\,\left (x+{\mathrm {e}}^5+2\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,\left (x+{\mathrm {e}}^5+1\right )}}{x^2+2\,x\,\ln \left (x+{\mathrm {e}}^5\right )+{\ln \left (x+{\mathrm {e}}^5\right )}^2}-6\,{\mathrm {e}}^x-3\,{\mathrm {e}}^{2\,x}+\frac {\frac {\left (x+{\mathrm {e}}^5\right )\,\left (x+2\,{\mathrm {e}}^5+{\mathrm {e}}^{10}+2\,x\,{\mathrm {e}}^5+x^2+1\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}-\frac {\ln \left (x+{\mathrm {e}}^5\right )\,\left (x+{\mathrm {e}}^5\right )}{2\,{\left (x+{\mathrm {e}}^5+1\right )}^3}}{x+\ln \left (x+{\mathrm {e}}^5\right )}+\frac {x+{\mathrm {e}}^5}{2\,x^3+\left (6\,{\mathrm {e}}^5+6\right )\,x^2+\left (12\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+6\right )\,x+6\,{\mathrm {e}}^5+6\,{\mathrm {e}}^{10}+2\,{\mathrm {e}}^{15}+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + x*exp(5) + log(x + exp(5))^2*(exp(x)*(18*x*exp(5) + 18*x^2) + exp(2*x)*(18*x*exp(5) + 18*x^2)) + e
xp(x)*(6*x^3*exp(5) + 6*x^4) - log(x + exp(5))*(x + exp(5) - exp(x)*(18*x^2*exp(5) + 18*x^3) - exp(2*x)*(18*x^
2*exp(5) + 18*x^3)) + log(x + exp(5))^3*(exp(x)*(6*x + 6*exp(5)) + exp(2*x)*(6*x + 6*exp(5))) + exp(2*x)*(6*x^
3*exp(5) + 6*x^4) + x^2)/(log(x + exp(5))^2*(3*x*exp(5) + 3*x^2) + log(x + exp(5))*(3*x^2*exp(5) + 3*x^3) + x^
3*exp(5) + log(x + exp(5))^3*(x + exp(5)) + x^4),x)

[Out]

((x*(x + exp(5) + 2))/(2*(x + exp(5) + 1)) - (log(x + exp(5))*(x + exp(5)))/(2*(x + exp(5) + 1)))/(log(x + exp
(5))^2 + 2*x*log(x + exp(5)) + x^2) - 6*exp(x) - 3*exp(2*x) + (((x + exp(5))*(x + 2*exp(5) + exp(10) + 2*x*exp
(5) + x^2 + 1))/(2*(x + exp(5) + 1)^3) - (log(x + exp(5))*(x + exp(5)))/(2*(x + exp(5) + 1)^3))/(x + log(x + e
xp(5))) + (x + exp(5))/(6*exp(5) + 6*exp(10) + 2*exp(15) + x^2*(6*exp(5) + 6) + x*(12*exp(5) + 6*exp(10) + 6)
+ 2*x^3 + 2)

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sympy [A]  time = 0.59, size = 34, normalized size = 1.55 \begin {gather*} \frac {x}{x^{2} + 2 x \log {\left (x + e^{5} \right )} + \log {\left (x + e^{5} \right )}^{2}} - 3 e^{2 x} - 6 e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)**2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x)**3+((-18*x*exp(5)-18*x**2)*exp(x)**
2+(-18*x*exp(5)-18*x**2)*exp(x))*ln(exp(5)+x)**2+((-18*x**2*exp(5)-18*x**3)*exp(x)**2+(-18*x**2*exp(5)-18*x**3
)*exp(x)+exp(5)+x)*ln(exp(5)+x)+(-6*x**3*exp(5)-6*x**4)*exp(x)**2+(-6*x**3*exp(5)-6*x**4)*exp(x)-x*exp(5)-x**2
-2*x)/((exp(5)+x)*ln(exp(5)+x)**3+(3*x*exp(5)+3*x**2)*ln(exp(5)+x)**2+(3*x**2*exp(5)+3*x**3)*ln(exp(5)+x)+x**3
*exp(5)+x**4),x)

[Out]

x/(x**2 + 2*x*log(x + exp(5)) + log(x + exp(5))**2) - 3*exp(2*x) - 6*exp(x)

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