3.41.57 \(\int \frac {-8 x^4-16 x^5-8 x^6+(16 x^4+40 x^5+24 x^6+e^{2+x} (-4 x^3-4 x^4)+e^3 (16 x^3+40 x^4+24 x^5)) \log (e^3+x)+e^x (e^5 (12 x^2+20 x^3+4 x^4)+e^2 (12 x^3+20 x^4+4 x^5)) \log ^2(e^3+x)+e^{2 x} (e^7 (2 x+2 x^2)+e^4 (2 x^2+2 x^3)) \log ^3(e^3+x)}{(e^3+x) \log ^3(e^3+x)} \, dx\)

Optimal. Leaf size=26 \[ \left (e^{2+x} x+\frac {2 x \left (x+x^2\right )}{\log \left (e^3+x\right )}\right )^2 \]

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Rubi [F]  time = 9.84, 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 {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x)*(-4*x^3 - 4*x^4) + E^3*(16*x^3 + 40*x^4 +
24*x^5))*Log[E^3 + x] + E^x*(E^5*(12*x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x]^2 +
E^(2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^3 + x)*Log[E^3 + x]^3),x]

[Out]

E^(4 + 2*x)*x^2 - 80*E^12*ExpIntegralEi[2*Log[E^3 + x]] - 96*E^6*(1 - E^3)^2*ExpIntegralEi[2*Log[E^3 + x]] + 6
4*E^9*(2 - E^3)*ExpIntegralEi[2*Log[E^3 + x]] + 16*E^6*(6 - 20*E^3 + 15*E^6)*ExpIntegralEi[2*Log[E^3 + x]] + 3
60*E^9*ExpIntegralEi[3*Log[E^3 + x]] + 144*E^3*(1 - E^3)^2*ExpIntegralEi[3*Log[E^3 + x]] - 216*E^6*(2 - E^3)*E
xpIntegralEi[3*Log[E^3 + x]] - 144*E^3*(1 - 5*E^3 + 5*E^6)*ExpIntegralEi[3*Log[E^3 + x]] - 640*E^6*ExpIntegral
Ei[4*Log[E^3 + x]] - 64*(1 - E^3)^2*ExpIntegralEi[4*Log[E^3 + x]] + 256*E^3*(2 - E^3)*ExpIntegralEi[4*Log[E^3
+ x]] + 64*(1 - 10*E^3 + 15*E^6)*ExpIntegralEi[4*Log[E^3 + x]] + 500*E^3*ExpIntegralEi[5*Log[E^3 + x]] + 200*(
1 - 3*E^3)*ExpIntegralEi[5*Log[E^3 + x]] - 100*(2 - E^3)*ExpIntegralEi[5*Log[E^3 + x]] + (4*E^12*(1 - E^3)^2)/
Log[E^3 + x]^2 - (4*E^9*(1 - E^3)^2*(E^3 + x))/Log[E^3 + x]^2 + (4*E^6*(1 - E^3)^2*x*(E^3 + x))/Log[E^3 + x]^2
- (4*E^3*(1 - E^3)^2*x^2*(E^3 + x))/Log[E^3 + x]^2 + (4*(1 - E^3)^2*x^3*(E^3 + x))/Log[E^3 + x]^2 + (4*(2 - E
^3)*x^4*(E^3 + x))/Log[E^3 + x]^2 + (4*x^5*(E^3 + x))/Log[E^3 + x]^2 + (8*E^9*(2 - 5*E^3 + 3*E^6)*(E^3 + x))/L
og[E^3 + x] + (16*(1 - E^3)^2*x^3*(E^3 + x))/Log[E^3 + x] + (16*E^3*(2 - E^3)*x^3*(E^3 + x))/Log[E^3 + x] + (2
0*E^3*x^4*(E^3 + x))/Log[E^3 + x] + (20*(2 - E^3)*x^4*(E^3 + x))/Log[E^3 + x] + (24*x^5*(E^3 + x))/Log[E^3 + x
] - (8*E^6*(6 - 20*E^3 + 15*E^6)*(E^3 + x)^2)/Log[E^3 + x] + (48*E^3*(1 - 5*E^3 + 5*E^6)*(E^3 + x)^3)/Log[E^3
+ x] - (16*(1 - 10*E^3 + 15*E^6)*(E^3 + x)^4)/Log[E^3 + x] - (40*(1 - 3*E^3)*(E^3 + x)^5)/Log[E^3 + x] - (24*(
E^3 + x)^6)/Log[E^3 + x] + 4*E^15*LogIntegral[E^3 + x] + 16*E^9*(1 - E^3)^2*LogIntegral[E^3 + x] - 4*E^12*(2 -
E^3)*LogIntegral[E^3 + x] - 8*E^9*(2 - 5*E^3 + 3*E^6)*LogIntegral[E^3 + x] - 4*E^6*(1 - E^3)*Defer[Int][E^(2
+ x)/Log[E^3 + x]^2, x] - 8*(1 - E^3)*Defer[Int][E^(8 + x)/Log[E^3 + x]^2, x] + 4*Defer[Int][E^(11 + x)/Log[E^
3 + x]^2, x] + 4*(1 - E^3)*Defer[Int][E^(11 + x)/((E^3 + x)*Log[E^3 + x]^2), x] + 12*(1 - E^3)*Defer[Int][(E^(
5 + x)*(E^3 + x))/Log[E^3 + x]^2, x] - 12*Defer[Int][(E^(8 + x)*(E^3 + x))/Log[E^3 + x]^2, x] - 4*(1 - E^3)*De
fer[Int][(E^(2 + x)*(E^3 + x)^2)/Log[E^3 + x]^2, x] + 12*Defer[Int][(E^(5 + x)*(E^3 + x)^2)/Log[E^3 + x]^2, x]
- 4*Defer[Int][(E^(2 + x)*(E^3 + x)^3)/Log[E^3 + x]^2, x] + 4*(3 - 5*E^3 + E^6)*Defer[Int][E^(8 + x)/Log[E^3
+ x], x] - 4*E^3*(6 - 15*E^3 + 4*E^6)*Defer[Int][(E^(2 + x)*(E^3 + x))/Log[E^3 + x], x] + 12*(1 - 5*E^3 + 2*E^
6)*Defer[Int][(E^(2 + x)*(E^3 + x)^2)/Log[E^3 + x], x] + 4*(5 - 4*E^3)*Defer[Int][(E^(2 + x)*(E^3 + x)^3)/Log[
E^3 + x], x] + 4*Defer[Int][(E^(2 + x)*(E^3 + x)^4)/Log[E^3 + x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 x \left (e^{4+2 x} (1+x)-\frac {4 x^3 (1+x)^2}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )}+\frac {2 x^2 (1+x) \left (-e^{2+x}+2 x (2+3 x)+e^3 (4+6 x)\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )}+\frac {2 e^{2+x} x \left (3+5 x+x^2\right )}{\log \left (e^3+x\right )}\right ) \, dx\\ &=2 \int x \left (e^{4+2 x} (1+x)-\frac {4 x^3 (1+x)^2}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )}+\frac {2 x^2 (1+x) \left (-e^{2+x}+2 x (2+3 x)+e^3 (4+6 x)\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )}+\frac {2 e^{2+x} x \left (3+5 x+x^2\right )}{\log \left (e^3+x\right )}\right ) \, dx\\ &=2 \int \left (e^{4+2 x} x (1+x)+\frac {4 x^3 (1+x) \left (-x-x^2+2 e^3 \log \left (e^3+x\right )+2 \left (1+\frac {3 e^3}{2}\right ) x \log \left (e^3+x\right )+3 x^2 \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )}+\frac {2 e^{2+x} x^2 \left (-x-x^2+3 e^3 \log \left (e^3+x\right )+3 \left (1+\frac {5 e^3}{3}\right ) x \log \left (e^3+x\right )+5 \left (1+\frac {e^3}{5}\right ) x^2 \log \left (e^3+x\right )+x^3 \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )}\right ) \, dx\\ &=2 \int e^{4+2 x} x (1+x) \, dx+4 \int \frac {e^{2+x} x^2 \left (-x-x^2+3 e^3 \log \left (e^3+x\right )+3 \left (1+\frac {5 e^3}{3}\right ) x \log \left (e^3+x\right )+5 \left (1+\frac {e^3}{5}\right ) x^2 \log \left (e^3+x\right )+x^3 \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )} \, dx+8 \int \frac {x^3 (1+x) \left (-x-x^2+2 e^3 \log \left (e^3+x\right )+2 \left (1+\frac {3 e^3}{2}\right ) x \log \left (e^3+x\right )+3 x^2 \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx\\ &=2 \int \left (e^{4+2 x} x+e^{4+2 x} x^2\right ) \, dx+4 \int \frac {e^{2+x} x^2 \left (-x (1+x)+\left (e^3+x\right ) \left (3+5 x+x^2\right ) \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )} \, dx+8 \int \frac {x^3 (1+x) \left (-x (1+x)+\left (e^3+x\right ) (2+3 x) \log \left (e^3+x\right )\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx\\ &=2 \int e^{4+2 x} x \, dx+2 \int e^{4+2 x} x^2 \, dx+4 \int \left (-\frac {e^{2+x} x^3 (1+x)}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )}+\frac {e^{2+x} x^2 \left (3+5 x+x^2\right )}{\log \left (e^3+x\right )}\right ) \, dx+8 \int \left (-\frac {x^4 (1+x)^2}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )}+\frac {x^3 (1+x) (2+3 x)}{\log ^2\left (e^3+x\right )}\right ) \, dx\\ &=e^{4+2 x} x+e^{4+2 x} x^2-2 \int e^{4+2 x} x \, dx-4 \int \frac {e^{2+x} x^3 (1+x)}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )} \, dx+4 \int \frac {e^{2+x} x^2 \left (3+5 x+x^2\right )}{\log \left (e^3+x\right )} \, dx-8 \int \frac {x^4 (1+x)^2}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx+8 \int \frac {x^3 (1+x) (2+3 x)}{\log ^2\left (e^3+x\right )} \, dx-\int e^{4+2 x} \, dx\\ &=-\frac {1}{2} e^{4+2 x}+e^{4+2 x} x^2-4 \int \left (\frac {e^{2+x} \left (e^6-e^9\right )}{\log ^2\left (e^3+x\right )}+\frac {e^{5+x} \left (-1+e^3\right ) x}{\log ^2\left (e^3+x\right )}-\frac {e^{2+x} \left (-1+e^3\right ) x^2}{\log ^2\left (e^3+x\right )}+\frac {e^{2+x} x^3}{\log ^2\left (e^3+x\right )}+\frac {e^{11+x} \left (-1+e^3\right )}{\left (e^3+x\right ) \log ^2\left (e^3+x\right )}\right ) \, dx+4 \int \left (\frac {e^{8+x} \left (3-5 e^3+e^6\right )}{\log \left (e^3+x\right )}+\frac {e^{2+x} \left (-6 e^3+15 e^6-4 e^9\right ) \left (e^3+x\right )}{\log \left (e^3+x\right )}+\frac {3 e^{2+x} \left (1-5 e^3+2 e^6\right ) \left (e^3+x\right )^2}{\log \left (e^3+x\right )}+\frac {e^{2+x} \left (5-4 e^3\right ) \left (e^3+x\right )^3}{\log \left (e^3+x\right )}+\frac {e^{2+x} \left (e^3+x\right )^4}{\log \left (e^3+x\right )}\right ) \, dx-8 \int \left (-\frac {e^9 \left (-1+e^3\right )^2}{\log ^3\left (e^3+x\right )}+\frac {e^6 \left (-1+e^3\right )^2 x}{\log ^3\left (e^3+x\right )}-\frac {e^3 \left (-1+e^3\right )^2 x^2}{\log ^3\left (e^3+x\right )}+\frac {\left (-1+e^3\right )^2 x^3}{\log ^3\left (e^3+x\right )}-\frac {\left (-2+e^3\right ) x^4}{\log ^3\left (e^3+x\right )}+\frac {x^5}{\log ^3\left (e^3+x\right )}+\frac {e^{12} \left (-1+e^3\right )^2}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )}\right ) \, dx+8 \int \left (-\frac {e^9 \left (2-5 e^3+3 e^6\right )}{\log ^2\left (e^3+x\right )}+\frac {e^6 \left (6-20 e^3+15 e^6\right ) \left (e^3+x\right )}{\log ^2\left (e^3+x\right )}-\frac {6 \left (e^3-5 e^6+5 e^9\right ) \left (e^3+x\right )^2}{\log ^2\left (e^3+x\right )}+\frac {2 \left (1-10 e^3+15 e^6\right ) \left (e^3+x\right )^3}{\log ^2\left (e^3+x\right )}-\frac {5 \left (-1+3 e^3\right ) \left (e^3+x\right )^4}{\log ^2\left (e^3+x\right )}+\frac {3 \left (e^3+x\right )^5}{\log ^2\left (e^3+x\right )}\right ) \, dx+\int e^{4+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 33, normalized size = 1.27 \begin {gather*} \frac {x^2 \left (2 x (1+x)+e^{2+x} \log \left (e^3+x\right )\right )^2}{\log ^2\left (e^3+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x)*(-4*x^3 - 4*x^4) + E^3*(16*x^3 + 40
*x^4 + 24*x^5))*Log[E^3 + x] + E^x*(E^5*(12*x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x
]^2 + E^(2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^3 + x)*Log[E^3 + x]^3),x]

[Out]

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

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fricas [B]  time = 0.62, size = 59, normalized size = 2.27 \begin {gather*} \frac {4 \, x^{6} + 8 \, x^{5} + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{2} + 4 \, x^{4} + 4 \, {\left (x^{4} + x^{3}\right )} e^{\left (x + 2\right )} \log \left (x + e^{3}\right )}{\log \left (x + e^{3}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="fricas")

[Out]

(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4 + x^3)*e^(x + 2)*log(x + e^3))/log(x + e^3)^2

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giac [B]  time = 0.35, size = 88, normalized size = 3.38 \begin {gather*} \frac {4 \, x^{6} \log \left (x + e^{3}\right ) + 4 \, x^{4} e^{\left (x + 2\right )} \log \left (x + e^{3}\right )^{2} + 8 \, x^{5} \log \left (x + e^{3}\right ) + 4 \, x^{3} e^{\left (x + 2\right )} \log \left (x + e^{3}\right )^{2} + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{3} + 4 \, x^{4} \log \left (x + e^{3}\right )}{\log \left (x + e^{3}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="giac")

[Out]

(4*x^6*log(x + e^3) + 4*x^4*e^(x + 2)*log(x + e^3)^2 + 8*x^5*log(x + e^3) + 4*x^3*e^(x + 2)*log(x + e^3)^2 + x
^2*e^(2*x + 4)*log(x + e^3)^3 + 4*x^4*log(x + e^3))/log(x + e^3)^3

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maple [B]  time = 0.12, size = 55, normalized size = 2.12

method	result	size
risch	\(x^{2} {\mathrm e}^{2 x +4}+\frac {4 x^{3} \left (\ln \left ({\mathrm e}^{3}+x \right ) x \,{\mathrm e}^{2+x}+\ln \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{2+x}+x^{3}+2 x^{2}+x \right )}{\ln \left ({\mathrm e}^{3}+x \right )^{2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*ln(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1
)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*ln(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5+40*x^
4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*ln(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/ln(exp(3)+x)^3,x,method=_RE
TURNVERBOSE)

[Out]

x^2*exp(2*x+4)+4*x^3*(ln(exp(3)+x)*x*exp(2+x)+ln(exp(3)+x)*exp(2+x)+x^3+2*x^2+x)/ln(exp(3)+x)^2

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maxima [B]  time = 0.41, size = 63, normalized size = 2.42 \begin {gather*} \frac {4 \, x^{6} + 8 \, x^{5} + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{2} + 4 \, x^{4} + 4 \, {\left (x^{4} e^{2} + x^{3} e^{2}\right )} e^{x} \log \left (x + e^{3}\right )}{\log \left (x + e^{3}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="maxima")

[Out]

(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4*e^2 + x^3*e^2)*e^x*log(x + e^3))/log(x + e^3)
^2

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mupad [B]  time = 4.06, size = 79, normalized size = 3.04 \begin {gather*} \frac {4\,x^4}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+\frac {8\,x^5}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+\frac {4\,x^6}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+x^2\,{\mathrm {e}}^{2\,x+4}+\frac {4\,x^3\,{\mathrm {e}}^{x+2}}{\ln \left (x+{\mathrm {e}}^3\right )}+\frac {4\,x^4\,{\mathrm {e}}^{x+2}}{\ln \left (x+{\mathrm {e}}^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(3))*(exp(3)*(16*x^3 + 40*x^4 + 24*x^5) + 16*x^4 + 40*x^5 + 24*x^6 - exp(2)*exp(x)*(4*x^3 + 4*
x^4)) - 8*x^4 - 16*x^5 - 8*x^6 + exp(x)*log(x + exp(3))^2*(exp(2)*(12*x^3 + 20*x^4 + 4*x^5) + exp(5)*(12*x^2 +
20*x^3 + 4*x^4)) + exp(2*x)*log(x + exp(3))^3*(exp(7)*(2*x + 2*x^2) + exp(4)*(2*x^2 + 2*x^3)))/(log(x + exp(3
))^3*(x + exp(3))),x)

[Out]

(4*x^4)/log(x + exp(3))^2 + (8*x^5)/log(x + exp(3))^2 + (4*x^6)/log(x + exp(3))^2 + x^2*exp(2*x + 4) + (4*x^3*
exp(x + 2))/log(x + exp(3)) + (4*x^4*exp(x + 2))/log(x + exp(3))

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sympy [B]  time = 0.47, size = 68, normalized size = 2.62 \begin {gather*} \frac {x^{2} e^{4} e^{2 x} \log {\left (x + e^{3} \right )} + \left (4 x^{4} e^{2} + 4 x^{3} e^{2}\right ) e^{x}}{\log {\left (x + e^{3} \right )}} + \frac {4 x^{6} + 8 x^{5} + 4 x^{4}}{\log {\left (x + e^{3} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2+2*x)*exp(1)**4*exp(3)+(2*x**3+2*x**2)*exp(1)**4)*exp(x)**2*ln(exp(3)+x)**3+((4*x**4+20*x**
3+12*x**2)*exp(1)**2*exp(3)+(4*x**5+20*x**4+12*x**3)*exp(1)**2)*exp(x)*ln(exp(3)+x)**2+((-4*x**4-4*x**3)*exp(1
)**2*exp(x)+(24*x**5+40*x**4+16*x**3)*exp(3)+24*x**6+40*x**5+16*x**4)*ln(exp(3)+x)-8*x**6-16*x**5-8*x**4)/(exp
(3)+x)/ln(exp(3)+x)**3,x)

[Out]

(x**2*exp(4)*exp(2*x)*log(x + exp(3)) + (4*x**4*exp(2) + 4*x**3*exp(2))*exp(x))/log(x + exp(3)) + (4*x**6 + 8*
x**5 + 4*x**4)/log(x + exp(3))**2

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