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3.52.55 \(\int \frac {162 e^x+e^{4 x^2} (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4)+e^{2 x^2} (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x (-27-27 x-36 x^2))}{162+e^{2 x^2} (72 e^{2 x}-144 e^x x+72 x^2)+e^{4 x^2} (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4)} \, dx\)

Optimal. Leaf size=30 \[ e^x+\frac {3}{-8-\frac {16}{9} e^{2 x^2} \left (e^x-x\right )^2} \]

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Rubi [F]  time = 8.87, 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 {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E^(2*x)*x^3 + 8*E^x*x^4) + E^(2*x^2)*
(72*E^(3*x) + E^(2*x)*(27 - 90*x) + 27*x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(2*x) -
 144*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x)*x^2 - 32*E^x*x^3 + 8*x^4)),x]

[Out]

81*Defer[Int][E^x/(-9 - 2*E^(2*x*(1 + x)) + 4*E^(x + 2*x^2)*x - 2*E^(2*x^2)*x^2)^2, x] - (27*Defer[Int][E^(x +
 2*x^2)/(-9 - 2*E^(2*x*(1 + x)) + 4*E^(x + 2*x^2)*x - 2*E^(2*x^2)*x^2)^2, x])/2 - (27*Defer[Int][(E^(x + 2*x^2
)*x)/(-9 - 2*E^(2*x*(1 + x)) + 4*E^(x + 2*x^2)*x - 2*E^(2*x^2)*x^2)^2, x])/2 - 18*Defer[Int][(E^(x + 2*x^2)*x^
2)/(-9 - 2*E^(2*x*(1 + x)) + 4*E^(x + 2*x^2)*x - 2*E^(2*x^2)*x^2)^2, x] + 4*Defer[Int][(E^(x + 4*x^2)*x^4)/(-9
 - 2*E^(2*x*(1 + x)) + 4*E^(x + 2*x^2)*x - 2*E^(2*x^2)*x^2)^2, x] + (27*Defer[Int][E^(2*x*(1 + x))/(9 + 2*E^(2
*x*(1 + x)) - 4*E^(x + 2*x^2)*x + 2*E^(2*x^2)*x^2)^2, x])/2 + 36*Defer[Int][E^(3*x + 2*x^2)/(9 + 2*E^(2*x*(1 +
 x)) - 4*E^(x + 2*x^2)*x + 2*E^(2*x^2)*x^2)^2, x] + 4*Defer[Int][E^(5*x + 4*x^2)/(9 + 2*E^(2*x*(1 + x)) - 4*E^
(x + 2*x^2)*x + 2*E^(2*x^2)*x^2)^2, x] + (27*Defer[Int][(E^(2*x^2)*x)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)
*x + 2*E^(2*x^2)*x^2)^2, x])/2 - 45*Defer[Int][(E^(2*x*(1 + x))*x)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)*x
+ 2*E^(2*x^2)*x^2)^2, x] - 16*Defer[Int][(E^(4*x*(1 + x))*x)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)*x + 2*E^
(2*x^2)*x^2)^2, x] + 24*Defer[Int][(E^(3*x + 4*x^2)*x^2)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)*x + 2*E^(2*x
^2)*x^2)^2, x] + 27*Defer[Int][(E^(2*x^2)*x^3)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)*x + 2*E^(2*x^2)*x^2)^2
, x] - 16*Defer[Int][(E^(2*x*(1 + 2*x))*x^3)/(9 + 2*E^(2*x*(1 + x)) - 4*E^(x + 2*x^2)*x + 2*E^(2*x^2)*x^2)^2,
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{2 \left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {162 e^x+8 e^{x+4 x^2} \left (e^x-x\right )^4+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)-9 e^x \left (3+3 x+4 x^2\right )+27 \left (x+2 x^3\right )\right )}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {162 e^x}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2}-\frac {27 e^{x+2 x^2}}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2}-\frac {27 e^{x+2 x^2} x}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2}-\frac {36 e^{x+2 x^2} x^2}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2}+\frac {8 e^{x+4 x^2} x^4}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2}+\frac {27 e^{2 x (1+x)}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}+\frac {72 e^{3 x+2 x^2}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}+\frac {8 e^{5 x+4 x^2}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}+\frac {27 e^{2 x^2} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}-\frac {90 e^{2 x (1+x)} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}-\frac {32 e^{4 x (1+x)} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}+\frac {48 e^{3 x+4 x^2} x^2}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}+\frac {54 e^{2 x^2} x^3}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}-\frac {32 e^{2 x (1+2 x)} x^3}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2}\right ) \, dx\\ &=4 \int \frac {e^{x+4 x^2} x^4}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2} \, dx+4 \int \frac {e^{5 x+4 x^2}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx-\frac {27}{2} \int \frac {e^{x+2 x^2}}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2} \, dx-\frac {27}{2} \int \frac {e^{x+2 x^2} x}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2} \, dx+\frac {27}{2} \int \frac {e^{2 x (1+x)}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx+\frac {27}{2} \int \frac {e^{2 x^2} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx-16 \int \frac {e^{4 x (1+x)} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx-16 \int \frac {e^{2 x (1+2 x)} x^3}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx-18 \int \frac {e^{x+2 x^2} x^2}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2} \, dx+24 \int \frac {e^{3 x+4 x^2} x^2}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx+27 \int \frac {e^{2 x^2} x^3}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx+36 \int \frac {e^{3 x+2 x^2}}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx-45 \int \frac {e^{2 x (1+x)} x}{\left (9+2 e^{2 x (1+x)}-4 e^{x+2 x^2} x+2 e^{2 x^2} x^2\right )^2} \, dx+81 \int \frac {e^x}{\left (-9-2 e^{2 x (1+x)}+4 e^{x+2 x^2} x-2 e^{2 x^2} x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 23.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E^(2*x)*x^3 + 8*E^x*x^4) + E^(2
*x^2)*(72*E^(3*x) + E^(2*x)*(27 - 90*x) + 27*x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(
2*x) - 144*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x)*x^2 - 32*E^x*x^3 + 8*x^4)),x]

[Out]

Integrate[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E^(2*x)*x^3 + 8*E^x*x^4) + E^(2
*x^2)*(72*E^(3*x) + E^(2*x)*(27 - 90*x) + 27*x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(
2*x) - 144*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x)*x^2 - 32*E^x*x^3 + 8*x^4)), x]

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fricas [B]  time = 0.59, size = 59, normalized size = 1.97 \begin {gather*} \frac {16 \, {\left (x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 72 \, e^{x} - 27}{8 \, {\left (2 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp(x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-9
0*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp(x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*ex
p(x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2)*exp(x^2)^2+162),x, algorithm="fri
cas")

[Out]

1/8*(16*(x^2*e^x - 2*x*e^(2*x) + e^(3*x))*e^(2*x^2) + 72*e^x - 27)/(2*(x^2 - 2*x*e^x + e^(2*x))*e^(2*x^2) + 9)

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giac [B]  time = 3.29, size = 3224, normalized size = 107.47 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp(x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-9
0*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp(x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*ex
p(x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2)*exp(x^2)^2+162),x, algorithm="gia
c")

[Out]

1/4*(64*x^8*e^(16*x^2 + x) + 2304*x^8*e^(14*x^2 + x) + 20736*x^8*e^(12*x^2 + x) - 256*x^7*e^(16*x^2 + 2*x) - 2
56*x^7*e^(16*x^2 + x) - 9216*x^7*e^(14*x^2 + 2*x) - 2304*x^7*e^(14*x^2 + x) - 82944*x^7*e^(12*x^2 + 2*x) + 414
72*x^7*e^(12*x^2 + x) - 216*x^6*e^(14*x^2) - 7776*x^6*e^(12*x^2) - 69984*x^6*e^(10*x^2) + 384*x^6*e^(16*x^2 +
3*x) + 1024*x^6*e^(16*x^2 + 2*x) + 384*x^6*e^(16*x^2 + x) + 13824*x^6*e^(14*x^2 + 3*x) + 9216*x^6*e^(14*x^2 +
2*x) - 1152*x^6*e^(14*x^2 + x) + 124416*x^6*e^(12*x^2 + 3*x) - 165888*x^6*e^(12*x^2 + 2*x) + 51840*x^6*e^(12*x
^2 + x) + 186624*x^6*e^(10*x^2 + x) + 864*x^5*e^(14*x^2) + 7776*x^5*e^(12*x^2) - 139968*x^5*e^(10*x^2) - 256*x
^5*e^(16*x^2 + 4*x) - 1536*x^5*e^(16*x^2 + 3*x) - 1536*x^5*e^(16*x^2 + 2*x) - 256*x^5*e^(16*x^2 + x) - 9216*x^
5*e^(14*x^2 + 4*x) - 13824*x^5*e^(14*x^2 + 3*x) + 5760*x^5*e^(14*x^2 + 2*x) - 720*x^5*e^(14*x^2 + x) - 82944*x
^5*e^(12*x^2 + 4*x) + 248832*x^5*e^(12*x^2 + 3*x) - 165888*x^5*e^(12*x^2 + 2*x) + 5184*x^5*e^(12*x^2 + x) - 37
3248*x^5*e^(10*x^2 + 2*x) + 513216*x^5*e^(10*x^2 + x) - 1296*x^4*e^(14*x^2) + 4860*x^4*e^(12*x^2) - 139968*x^4
*e^(10*x^2) - 314928*x^4*e^(8*x^2) + 64*x^4*e^(16*x^2 + 5*x) + 1024*x^4*e^(16*x^2 + 4*x) + 2304*x^4*e^(16*x^2
+ 3*x) + 1024*x^4*e^(16*x^2 + 2*x) + 64*x^4*e^(16*x^2 + x) + 2304*x^4*e^(14*x^2 + 5*x) + 9216*x^4*e^(14*x^2 +
4*x) - 9792*x^4*e^(14*x^2 + 3*x) - 216*x^4*e^(14*x^2 + 2*x) + 2304*x^4*e^(14*x^2 + x) + 20736*x^4*e^(12*x^2 +
5*x) - 165888*x^4*e^(12*x^2 + 4*x) + 207360*x^4*e^(12*x^2 + 3*x) - 7776*x^4*e^(12*x^2 + 2*x) - 28512*x^4*e^(12
*x^2 + x) + 186624*x^4*e^(10*x^2 + 3*x) - 816480*x^4*e^(10*x^2 + 2*x) + 606528*x^4*e^(10*x^2 + x) + 419904*x^4
*e^(8*x^2 + x) + 864*x^3*e^(14*x^2) - 629856*x^3*e^(8*x^2) - 256*x^3*e^(16*x^2 + 5*x) - 1536*x^3*e^(16*x^2 + 4
*x) - 1536*x^3*e^(16*x^2 + 3*x) - 256*x^3*e^(16*x^2 + 2*x) - 2304*x^3*e^(14*x^2 + 5*x) + 6912*x^3*e^(14*x^2 +
4*x) + 4608*x^3*e^(14*x^2 + 3*x) - 8352*x^3*e^(14*x^2 + 2*x) + 288*x^3*e^(14*x^2 + x) + 41472*x^3*e^(12*x^2 +
5*x) - 124416*x^3*e^(12*x^2 + 4*x) + 41472*x^3*e^(12*x^2 + 3*x) + 33696*x^3*e^(12*x^2 + 2*x) - 6480*x^3*e^(12*
x^2 + x) + 373248*x^3*e^(10*x^2 + 3*x) - 699840*x^3*e^(10*x^2 + 2*x) + 256608*x^3*e^(10*x^2 + x) + 839808*x^3*
e^(8*x^2 + x) - 216*x^2*e^(14*x^2) - 7776*x^2*e^(12*x^2) + 21870*x^2*e^(10*x^2) - 472392*x^2*e^(8*x^2) + 384*x
^2*e^(16*x^2 + 5*x) + 1024*x^2*e^(16*x^2 + 4*x) + 384*x^2*e^(16*x^2 + 3*x) - 1728*x^2*e^(14*x^2 + 5*x) - 4608*
x^2*e^(14*x^2 + 4*x) + 6912*x^2*e^(14*x^2 + 3*x) + 3312*x^2*e^(14*x^2 + 2*x) - 1152*x^2*e^(14*x^2 + x) + 31104
*x^2*e^(12*x^2 + 5*x) - 41472*x^2*e^(12*x^2 + 4*x) - 7776*x^2*e^(12*x^2 + 3*x) - 14904*x^2*e^(12*x^2 + 2*x) +
20736*x^2*e^(12*x^2 + x) + 279936*x^2*e^(10*x^2 + 3*x) - 291600*x^2*e^(10*x^2 + 2*x) + 46656*x^2*e^(10*x^2 + x
) + 629856*x^2*e^(8*x^2 + x) + 3888*x*e^(12*x^2) - 17496*x*e^(10*x^2) - 157464*x*e^(8*x^2) - 256*x*e^(16*x^2 +
 5*x) - 256*x*e^(16*x^2 + 4*x) + 1152*x*e^(14*x^2 + 5*x) - 2304*x*e^(14*x^2 + 4*x) - 2304*x*e^(14*x^2 + 3*x) -
 288*x*e^(14*x^2 + 2*x) + 432*x*e^(14*x^2 + x) + 10368*x*e^(12*x^2 + 5*x) - 5184*x*e^(12*x^2 + 4*x) + 10368*x*
e^(12*x^2 + 3*x) - 14256*x*e^(12*x^2 + 2*x) - 1296*x*e^(12*x^2 + x) + 93312*x*e^(10*x^2 + 3*x) - 58320*x*e^(10
*x^2 + 2*x) + 32076*x*e^(10*x^2 + x) + 209952*x*e^(8*x^2 + x) - 972*e^(12*x^2) - 8748*e^(10*x^2) - 19683*e^(8*
x^2) + 64*e^(16*x^2 + 5*x) + 576*e^(14*x^2 + 5*x) + 576*e^(14*x^2 + 3*x) - 216*e^(14*x^2 + 2*x) + 1296*e^(12*x
^2 + 5*x) + 5184*e^(12*x^2 + 3*x) - 1944*e^(12*x^2 + 2*x) + 1296*e^(12*x^2 + x) + 11664*e^(10*x^2 + 3*x) - 437
4*e^(10*x^2 + 2*x) + 11664*e^(10*x^2 + x) + 26244*e^(8*x^2 + x))/(16*x^8*e^(16*x^2) + 576*x^8*e^(14*x^2) + 518
4*x^8*e^(12*x^2) - 64*x^7*e^(16*x^2) - 576*x^7*e^(14*x^2) + 10368*x^7*e^(12*x^2) - 64*x^7*e^(16*x^2 + x) - 230
4*x^7*e^(14*x^2 + x) - 20736*x^7*e^(12*x^2 + x) + 96*x^6*e^(16*x^2) - 288*x^6*e^(14*x^2) + 12960*x^6*e^(12*x^2
) + 46656*x^6*e^(10*x^2) + 96*x^6*e^(16*x^2 + 2*x) + 256*x^6*e^(16*x^2 + x) + 3456*x^6*e^(14*x^2 + 2*x) + 2304
*x^6*e^(14*x^2 + x) + 31104*x^6*e^(12*x^2 + 2*x) - 41472*x^6*e^(12*x^2 + x) - 64*x^5*e^(16*x^2) - 288*x^5*e^(1
4*x^2) - 2592*x^5*e^(12*x^2) + 93312*x^5*e^(10*x^2) - 64*x^5*e^(16*x^2 + 3*x) - 384*x^5*e^(16*x^2 + 2*x) - 384
*x^5*e^(16*x^2 + x) - 2304*x^5*e^(14*x^2 + 3*x) - 3456*x^5*e^(14*x^2 + 2*x) + 1440*x^5*e^(14*x^2 + x) - 20736*
x^5*e^(12*x^2 + 3*x) + 62208*x^5*e^(12*x^2 + 2*x) - 41472*x^5*e^(12*x^2 + x) - 93312*x^5*e^(10*x^2 + x) + 16*x
^4*e^(16*x^2) + 1008*x^4*e^(14*x^2) - 3240*x^4*e^(12*x^2) + 81648*x^4*e^(10*x^2) + 104976*x^4*e^(8*x^2) + 16*x
^4*e^(16*x^2 + 4*x) + 256*x^4*e^(16*x^2 + 3*x) + 576*x^4*e^(16*x^2 + 2*x) + 256*x^4*e^(16*x^2 + x) + 576*x^4*e
^(14*x^2 + 4*x) + 2304*x^4*e^(14*x^2 + 3*x) - 2448*x^4*e^(14*x^2 + 2*x) + 5184*x^4*e^(12*x^2 + 4*x) - 41472*x^
4*e^(12*x^2 + 3*x) + 51840*x^4*e^(12*x^2 + 2*x) + 46656*x^4*e^(10*x^2 + 2*x) - 186624*x^4*e^(10*x^2 + x) - 576
*x^3*e^(14*x^2) + 1296*x^3*e^(12*x^2) + 11664*x^3*e^(10*x^2) + 209952*x^3*e^(8*x^2) - 64*x^3*e^(16*x^2 + 4*x)
- 384*x^3*e^(16*x^2 + 3*x) - 384*x^3*e^(16*x^2 + 2*x) - 64*x^3*e^(16*x^2 + x) - 576*x^3*e^(14*x^2 + 4*x) + 172
8*x^3*e^(14*x^2 + 3*x) + 1152*x^3*e^(14*x^2 + 2*x) - 2304*x^3*e^(14*x^2 + x) + 10368*x^3*e^(12*x^2 + 4*x) - 31
104*x^3*e^(12*x^2 + 3*x) + 10368*x^3*e^(12*x^2 + 2*x) + 6480*x^3*e^(12*x^2 + x) + 93312*x^3*e^(10*x^2 + 2*x) -
 139968*x^3*e^(10*x^2 + x) + 144*x^2*e^(14*x^2) + 3240*x^2*e^(12*x^2) - 5832*x^2*e^(10*x^2) + 157464*x^2*e^(8*
x^2) + 96*x^2*e^(16*x^2 + 4*x) + 256*x^2*e^(16*x^2 + 3*x) + 96*x^2*e^(16*x^2 + 2*x) - 432*x^2*e^(14*x^2 + 4*x)
 - 1152*x^2*e^(14*x^2 + 3*x) + 1728*x^2*e^(14*x^2 + 2*x) + 1152*x^2*e^(14*x^2 + x) + 7776*x^2*e^(12*x^2 + 4*x)
 - 10368*x^2*e^(12*x^2 + 3*x) - 1944*x^2*e^(12*x^2 + 2*x) - 5184*x^2*e^(12*x^2 + x) + 69984*x^2*e^(10*x^2 + 2*
x) - 46656*x^2*e^(10*x^2 + x) - 1296*x*e^(12*x^2) + 5832*x*e^(10*x^2) + 52488*x*e^(8*x^2) - 64*x*e^(16*x^2 + 4
*x) - 64*x*e^(16*x^2 + 3*x) + 288*x*e^(14*x^2 + 4*x) - 576*x*e^(14*x^2 + 3*x) - 576*x*e^(14*x^2 + 2*x) - 288*x
*e^(14*x^2 + x) + 2592*x*e^(12*x^2 + 4*x) - 1296*x*e^(12*x^2 + 3*x) + 2592*x*e^(12*x^2 + 2*x) - 2592*x*e^(12*x
^2 + x) + 23328*x*e^(10*x^2 + 2*x) - 5832*x*e^(10*x^2 + x) + 324*e^(12*x^2) + 2916*e^(10*x^2) + 6561*e^(8*x^2)
 + 16*e^(16*x^2 + 4*x) + 144*e^(14*x^2 + 4*x) + 144*e^(14*x^2 + 2*x) + 324*e^(12*x^2 + 4*x) + 1296*e^(12*x^2 +
 2*x) + 2916*e^(10*x^2 + 2*x))

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maple [A]  time = 0.08, size = 41, normalized size = 1.37

method result size
risch \({\mathrm e}^{x}-\frac {27}{8 \left (2 \,{\mathrm e}^{2 \left (x +1\right ) x}-4 x \,{\mathrm e}^{\left (2 x +1\right ) x}+2 x^{2} {\mathrm e}^{2 x^{2}}+9\right )}\) \(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp(x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-90*x+27
)*exp(x)^2+(-36*x^2-27*x-27)*exp(x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*exp(x)^2
*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2)*exp(x^2)^2+162),x,method=_RETURNVERBOSE
)

[Out]

exp(x)-27/8/(2*exp(2*(x+1)*x)-4*x*exp((2*x+1)*x)+2*x^2*exp(2*x^2)+9)

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maxima [B]  time = 0.49, size = 59, normalized size = 1.97 \begin {gather*} \frac {16 \, {\left (x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 72 \, e^{x} - 27}{8 \, {\left (2 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp(x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-9
0*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp(x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*ex
p(x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2)*exp(x^2)^2+162),x, algorithm="max
ima")

[Out]

1/8*(16*(x^2*e^x - 2*x*e^(2*x) + e^(3*x))*e^(2*x^2) + 72*e^x - 27)/(2*(x^2 - 2*x*e^x + e^(2*x))*e^(2*x^2) + 9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {162\,{\mathrm {e}}^x+{\mathrm {e}}^{4\,x^2}\,\left (8\,{\mathrm {e}}^{5\,x}-32\,x\,{\mathrm {e}}^{4\,x}+8\,x^4\,{\mathrm {e}}^x+48\,x^2\,{\mathrm {e}}^{3\,x}-32\,x^3\,{\mathrm {e}}^{2\,x}\right )+{\mathrm {e}}^{2\,x^2}\,\left (27\,x+72\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x\,\left (36\,x^2+27\,x+27\right )-{\mathrm {e}}^{2\,x}\,\left (90\,x-27\right )+54\,x^3\right )}{{\mathrm {e}}^{4\,x^2}\,\left (8\,{\mathrm {e}}^{4\,x}-32\,x\,{\mathrm {e}}^{3\,x}-32\,x^3\,{\mathrm {e}}^x+48\,x^2\,{\mathrm {e}}^{2\,x}+8\,x^4\right )+{\mathrm {e}}^{2\,x^2}\,\left (72\,{\mathrm {e}}^{2\,x}-144\,x\,{\mathrm {e}}^x+72\,x^2\right )+162} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((162*exp(x) + exp(4*x^2)*(8*exp(5*x) - 32*x*exp(4*x) + 8*x^4*exp(x) + 48*x^2*exp(3*x) - 32*x^3*exp(2*x)) +
 exp(2*x^2)*(27*x + 72*exp(3*x) - exp(x)*(27*x + 36*x^2 + 27) - exp(2*x)*(90*x - 27) + 54*x^3))/(exp(4*x^2)*(8
*exp(4*x) - 32*x*exp(3*x) - 32*x^3*exp(x) + 48*x^2*exp(2*x) + 8*x^4) + exp(2*x^2)*(72*exp(2*x) - 144*x*exp(x)
+ 72*x^2) + 162),x)

[Out]

int((162*exp(x) + exp(4*x^2)*(8*exp(5*x) - 32*x*exp(4*x) + 8*x^4*exp(x) + 48*x^2*exp(3*x) - 32*x^3*exp(2*x)) +
 exp(2*x^2)*(27*x + 72*exp(3*x) - exp(x)*(27*x + 36*x^2 + 27) - exp(2*x)*(90*x - 27) + 54*x^3))/(exp(4*x^2)*(8
*exp(4*x) - 32*x*exp(3*x) - 32*x^3*exp(x) + 48*x^2*exp(2*x) + 8*x^4) + exp(2*x^2)*(72*exp(2*x) - 144*x*exp(x)
+ 72*x^2) + 162), x)

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sympy [A]  time = 0.39, size = 31, normalized size = 1.03 \begin {gather*} e^{x} - \frac {27}{\left (16 x^{2} - 32 x e^{x} + 16 e^{2 x}\right ) e^{2 x^{2}} + 72} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)**5-32*x*exp(x)**4+48*x**2*exp(x)**3-32*exp(x)**2*x**3+8*exp(x)*x**4)*exp(x**2)**4+(72*exp
(x)**3+(-90*x+27)*exp(x)**2+(-36*x**2-27*x-27)*exp(x)+54*x**3+27*x)*exp(x**2)**2+162*exp(x))/((8*exp(x)**4-32*
x*exp(x)**3+48*exp(x)**2*x**2-32*exp(x)*x**3+8*x**4)*exp(x**2)**4+(72*exp(x)**2-144*exp(x)*x+72*x**2)*exp(x**2
)**2+162),x)

[Out]

exp(x) - 27/((16*x**2 - 32*x*exp(x) + 16*exp(2*x))*exp(2*x**2) + 72)

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