∫ −2ex8+2exx8+2x9−2x8 log(x)+(x7−8x8−9x9+e(8x7+8x8)+ex(−8x7−9x8−x9)+(8x7+8x8) log(x)) log(2+2x)______________________________________________________________________________

3.48.52 \(\int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+(x^7-8 x^8-9 x^9+e (8 x^7+8 x^8)+e^x (-8 x^7-9 x^8-x^9)+(8 x^7+8 x^8) \log (x)) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^8 \left (e-e^x-x+\log (x)\right )}{\log ^2(2 (1+x))} \]

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Rubi [F] time = 45.61, 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 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-2*E*x^8 + 2*E^x*x^8 + 2*x^9 - 2*x^8*Log[x] + (x^7 - 8*x^8 - 9*x^9 + E*(8*x^7 + 8*x^8) + E^x*(-8*x^7 - 9*

x^8 - x^9) + (8*x^7 + 8*x^8)*Log[x])*Log[2 + 2*x])/((1 + x)*Log[2 + 2*x]^3),x]

[Out]

126*ExpIntegralEi[2*Log[2*(1 + x)]] - 112*(1 - E)*ExpIntegralEi[2*Log[2*(1 + x)]] - 28*E*ExpIntegralEi[2*Log[2

*(1 + x)]] - (21*(1 + 8*E)*ExpIntegralEi[2*Log[2*(1 + x)]])/2 - 189*ExpIntegralEi[3*Log[2*(1 + x)]] + 168*(1 -

E)*ExpIntegralEi[3*Log[2*(1 + x)]] + 63*E*ExpIntegralEi[3*Log[2*(1 + x)]] + (105*(1 + 8*E)*ExpIntegralEi[3*Lo

g[2*(1 + x)]])/8 + (315*ExpIntegralEi[4*Log[2*(1 + x)]])/2 - 140*(1 - E)*ExpIntegralEi[4*Log[2*(1 + x)]] - 70*

E*ExpIntegralEi[4*Log[2*(1 + x)]] - (35*(1 + 8*E)*ExpIntegralEi[4*Log[2*(1 + x)]])/4 - (315*ExpIntegralEi[5*Lo

g[2*(1 + x)]])/4 + 70*(1 - E)*ExpIntegralEi[5*Log[2*(1 + x)]] + (175*E*ExpIntegralEi[5*Log[2*(1 + x)]])/4 + (1

05*(1 + 8*E)*ExpIntegralEi[5*Log[2*(1 + x)]])/32 + (189*ExpIntegralEi[6*Log[2*(1 + x)]])/8 - 21*(1 - E)*ExpInt

egralEi[6*Log[2*(1 + x)]] - (63*E*ExpIntegralEi[6*Log[2*(1 + x)]])/4 - (21*(1 + 8*E)*ExpIntegralEi[6*Log[2*(1

+ x)]])/32 - (63*ExpIntegralEi[7*Log[2*(1 + x)]])/16 + (7*(1 - E)*ExpIntegralEi[7*Log[2*(1 + x)]])/2 + (49*E*E

xpIntegralEi[7*Log[2*(1 + x)]])/16 + (7*(1 + 8*E)*ExpIntegralEi[7*Log[2*(1 + x)]])/128 + (9*ExpIntegralEi[8*Lo

g[2*(1 + x)]])/32 - ((1 - E)*ExpIntegralEi[8*Log[2*(1 + x)]])/4 - (E*ExpIntegralEi[8*Log[2*(1 + x)]])/4 + Log[

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

/Log[2*(1 + x)]^2 - 9/Log[2*(1 + x)] + (8*(1 - E))/Log[2*(1 + x)] + (1 + 8*E)/Log[2*(1 + x)] + (16*(1 + x))/Lo

g[2*(1 + x)] - (8*(1 - E)*(1 + x))/Log[2*(1 + x)] - ((1 + 8*E)*(1 + x))/Log[2*(1 + x)] - (7*(1 + x)^2)/Log[2*(

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

(1 + x))/Log[2 + 2*x]^2 + (E*x^3*(1 + x))/Log[2 + 2*x]^2 - (x^4*(1 + x))/Log[2 + 2*x]^2 - (E*x^4*(1 + x))/Log[

2 + 2*x]^2 + (x^5*(1 + x))/Log[2 + 2*x]^2 + (E*x^5*(1 + x))/Log[2 + 2*x]^2 - (x^6*(1 + x))/Log[2 + 2*x]^2 - (E

*x^6*(1 + x))/Log[2 + 2*x]^2 + (x^7*(1 + x))/Log[2 + 2*x]^2 + (E*x^7*(1 + x))/Log[2 + 2*x]^2 - (x^8*(1 + x))/L

og[2 + 2*x]^2 - (2*x*(1 + x))/Log[2 + 2*x] + (8*(1 - E)*x*(1 + x))/Log[2 + 2*x] + ((1 + 8*E)*x*(1 + x))/Log[2

+ 2*x] + (9*x^2*(1 + x))/Log[2 + 2*x] - (8*(1 - E)*x^2*(1 + x))/Log[2 + 2*x] - ((1 + 8*E)*x^2*(1 + x))/Log[2 +

2*x] - (9*x^3*(1 + x))/Log[2 + 2*x] + (8*(1 - E)*x^3*(1 + x))/Log[2 + 2*x] + ((1 + 8*E)*x^3*(1 + x))/Log[2 +

2*x] + (9*x^4*(1 + x))/Log[2 + 2*x] - (8*(1 - E)*x^4*(1 + x))/Log[2 + 2*x] - ((1 + 8*E)*x^4*(1 + x))/Log[2 + 2

*x] - (9*x^5*(1 + x))/Log[2 + 2*x] + (8*(1 - E)*x^5*(1 + x))/Log[2 + 2*x] + ((1 + 8*E)*x^5*(1 + x))/Log[2 + 2*

x] + (9*x^6*(1 + x))/Log[2 + 2*x] - (8*(1 - E)*x^6*(1 + x))/Log[2 + 2*x] - ((1 + 8*E)*x^6*(1 + x))/Log[2 + 2*x

] - (9*x^7*(1 + x))/Log[2 + 2*x] + (8*(1 - E)*x^7*(1 + x))/Log[2 + 2*x] + (8*E*x^7*(1 + x))/Log[2 + 2*x] - 36*

LogIntegral[2*(1 + x)] + 32*(1 - E)*LogIntegral[2*(1 + x)] + 4*E*LogIntegral[2*(1 + x)] + (7*(1 + 8*E)*LogInte

gral[2*(1 + x)])/2 - 16*Defer[Int][E^x/Log[2 + 2*x]^3, x] + 2*Defer[Int][E^x/((1 + x)*Log[2 + 2*x]^3), x] + 28

*Defer[Int][(E^x*(2 + 2*x))/Log[2 + 2*x]^3, x] - 28*Defer[Int][(E^x*(2 + 2*x)^2)/Log[2 + 2*x]^3, x] + (35*Defe

r[Int][(E^x*(2 + 2*x)^3)/Log[2 + 2*x]^3, x])/2 - 7*Defer[Int][(E^x*(2 + 2*x)^4)/Log[2 + 2*x]^3, x] + (7*Defer[

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

(E^x*(2 + 2*x)^7)/Log[2 + 2*x]^3, x]/64 + 2*Defer[Int][Log[x]/Log[2 + 2*x]^3, x] - 2*Defer[Int][(x*Log[x])/Log

[2 + 2*x]^3, x] + 2*Defer[Int][(x^2*Log[x])/Log[2 + 2*x]^3, x] - 2*Defer[Int][(x^3*Log[x])/Log[2 + 2*x]^3, x]

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

x^6*Log[x])/Log[2 + 2*x]^3, x] - 2*Defer[Int][(x^7*Log[x])/Log[2 + 2*x]^3, x] + 7*Defer[Int][E^x/Log[2 + 2*x]^

2, x] + 9*Defer[Int][E^x/((-1 - x)*Log[2 + 2*x]^2), x] + 9*Defer[Int][E^x/((1 + x)*Log[2 + 2*x]^2), x] - 24*De

fer[Int][(E^x*(2 + 2*x))/Log[2 + 2*x]^2, x] + 35*Defer[Int][(E^x*(2 + 2*x)^2)/Log[2 + 2*x]^2, x] - 28*Defer[In

t][(E^x*(2 + 2*x)^3)/Log[2 + 2*x]^2, x] + (105*Defer[Int][(E^x*(2 + 2*x)^4)/Log[2 + 2*x]^2, x])/8 - (7*Defer[I

nt][(E^x*(2 + 2*x)^5)/Log[2 + 2*x]^2, x])/2 + (7*Defer[Int][(E^x*(2 + 2*x)^6)/Log[2 + 2*x]^2, x])/16 - Defer[I

nt][(E^x*(2 + 2*x)^8)/Log[2 + 2*x]^2, x]/256 + 8*Defer[Int][(x^7*Log[x])/Log[2 + 2*x]^2, x] - 2*Defer[Subst][D

efer[Int][Log[-1 + x/2]/(x*Log[x]^3), x], x, 2 + 2*x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^7 \left (2 x \left (-e+e^x+x\right )-(1+x) \left (-1-8 e+9 x+e^x (8+x)\right ) \log (2 (1+x))-\log (x) (2 x-8 (1+x) \log (2 (1+x)))\right )}{(1+x) \log ^3(2+2 x)} \, dx\\ &=\int \left (\frac {e^x x^7 \left (2 x-8 \log (2 (1+x))-9 x \log (2 (1+x))-x^2 \log (2 (1+x))\right )}{(1+x) \log ^3(2+2 x)}+\frac {x^7 \left (-2 e x+2 x^2-2 x \log (x)+(1+8 e) \log (2 (1+x))-8 (1-e) x \log (2 (1+x))-9 x^2 \log (2 (1+x))+8 \log (x) \log (2 (1+x))+8 x \log (x) \log (2 (1+x))\right )}{(1+x) \log ^3(2+2 x)}\right ) \, dx\\ &=\int \frac {e^x x^7 \left (2 x-8 \log (2 (1+x))-9 x \log (2 (1+x))-x^2 \log (2 (1+x))\right )}{(1+x) \log ^3(2+2 x)} \, dx+\int \frac {x^7 \left (-2 e x+2 x^2-2 x \log (x)+(1+8 e) \log (2 (1+x))-8 (1-e) x \log (2 (1+x))-9 x^2 \log (2 (1+x))+8 \log (x) \log (2 (1+x))+8 x \log (x) \log (2 (1+x))\right )}{(1+x) \log ^3(2+2 x)} \, dx\\ &=\int \frac {e^x x^7 \left (2 x-\left (8+9 x+x^2\right ) \log (2 (1+x))\right )}{(1+x) \log ^3(2+2 x)} \, dx+\int \frac {x^7 (-2 (e-x) x+(1+8 e-9 x) (1+x) \log (2 (1+x))-\log (x) (2 x-8 (1+x) \log (2 (1+x))))}{(1+x) \log ^3(2+2 x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 2.39, size = 25, normalized size = 1.04 \begin {gather*} -\frac {x^8 \left (-e+e^x+x-\log (x)\right )}{\log ^2(2 (1+x))} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-2*E*x^8 + 2*E^x*x^8 + 2*x^9 - 2*x^8*Log[x] + (x^7 - 8*x^8 - 9*x^9 + E*(8*x^7 + 8*x^8) + E^x*(-8*x^

7 - 9*x^8 - x^9) + (8*x^7 + 8*x^8)*Log[x])*Log[2 + 2*x])/((1 + x)*Log[2 + 2*x]^3),x]

[Out]

-((x^8*(-E + E^x + x - Log[x]))/Log[2*(1 + x)]^2)

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fricas [A] time = 0.57, size = 34, normalized size = 1.42 \begin {gather*} -\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \relax (x)}{\log \left (2 \, x + 2\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)-9*x^9-8*x^8+x^7)*log(2*x+2)-2*

x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/(x+1)/log(2*x+2)^3,x, algorithm="fricas")

[Out]

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/log(2*x + 2)^2

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giac [A] time = 0.16, size = 34, normalized size = 1.42 \begin {gather*} -\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \relax (x)}{\log \left (2 \, x + 2\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)-9*x^9-8*x^8+x^7)*log(2*x+2)-2*

x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/(x+1)/log(2*x+2)^3,x, algorithm="giac")

[Out]

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/log(2*x + 2)^2

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maple [A] time = 0.09, size = 25, normalized size = 1.04


method	result	size

risch	\(\frac {\left (\ln \relax (x )+{\mathrm e}-{\mathrm e}^{x}-x \right ) x^{8}}{\ln \left (2 x +2\right )^{2}}\)	\(25\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^8+8*x^7)*ln(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)-9*x^9-8*x^8+x^7)*ln(2*x+2)-2*x^8*ln(x

)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/(x+1)/ln(2*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

(ln(x)+exp(1)-exp(x)-x)/ln(2*x+2)^2*x^8

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maxima [A] time = 0.50, size = 47, normalized size = 1.96 \begin {gather*} -\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \relax (x)}{\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (x + 1\right ) + \log \left (x + 1\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)-9*x^9-8*x^8+x^7)*log(2*x+2)-2*

x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/(x+1)/log(2*x+2)^3,x, algorithm="maxima")

[Out]

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/(log(2)^2 + 2*log(2)*log(x + 1) + log(x + 1)^2)

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mupad [B] time = 3.70, size = 57, normalized size = 2.38 \begin {gather*} \frac {x^8\,\ln \relax (x)}{{\ln \left (2\,x+2\right )}^2}-\frac {x^8\,{\mathrm {e}}^x}{{\ln \left (2\,x+2\right )}^2}-\frac {x^9}{{\ln \left (2\,x+2\right )}^2}+\frac {x^8\,\mathrm {e}}{{\ln \left (2\,x+2\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8*exp(x) - 2*x^8*log(x) + log(2*x + 2)*(log(x)*(8*x^7 + 8*x^8) + exp(1)*(8*x^7 + 8*x^8) + x^7 - 8*x^8

- 9*x^9 - exp(x)*(8*x^7 + 9*x^8 + x^9)) - 2*x^8*exp(1) + 2*x^9)/(log(2*x + 2)^3*(x + 1)),x)

[Out]

(x^8*log(x))/log(2*x + 2)^2 - (x^8*exp(x))/log(2*x + 2)^2 - x^9/log(2*x + 2)^2 + (x^8*exp(1))/log(2*x + 2)^2

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sympy [A] time = 0.42, size = 39, normalized size = 1.62 \begin {gather*} - \frac {x^{8} e^{x}}{\log {\left (2 x + 2 \right )}^{2}} + \frac {- x^{9} + x^{8} \log {\relax (x )} + e x^{8}}{\log {\left (2 x + 2 \right )}^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**8+8*x**7)*ln(x)+(-x**9-9*x**8-8*x**7)*exp(x)+(8*x**8+8*x**7)*exp(1)-9*x**9-8*x**8+x**7)*ln(2

*x+2)-2*x**8*ln(x)+2*x**8*exp(x)-2*x**8*exp(1)+2*x**9)/(x+1)/ln(2*x+2)**3,x)

[Out]

-x**8*exp(x)/log(2*x + 2)**2 + (-x**9 + x**8*log(x) + E*x**8)/log(2*x + 2)**2

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