3.51.75 \(\int (1+e^{16}+e^{20}+2 x-6 x^2+5 x^4+e^8 (-2+6 x^2)+e^{10} (-2+2 e^8+6 x^2)) \, dx\)

Optimal. Leaf size=19 \[ x^2+x \left (-1+e^8+e^{10}+x^2\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 3.00, number of steps used = 3, number of rules used = 0, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} x^5+2 e^{10} x^3+2 e^8 x^3-2 x^3+x^2+\left (1+e^{16}+e^{20}\right ) x-2 e^{10} \left (1-e^8\right ) x-2 e^8 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^16 + E^20 + 2*x - 6*x^2 + 5*x^4 + E^8*(-2 + 6*x^2) + E^10*(-2 + 2*E^8 + 6*x^2),x]

[Out]

-2*E^8*x - 2*E^10*(1 - E^8)*x + (1 + E^16 + E^20)*x + x^2 - 2*x^3 + 2*E^8*x^3 + 2*E^10*x^3 + x^5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (1+e^{16}+e^{20}\right ) x+x^2-2 x^3+x^5+e^8 \int \left (-2+6 x^2\right ) \, dx+e^{10} \int \left (-2+2 e^8+6 x^2\right ) \, dx\\ &=-2 e^8 x-2 e^{10} \left (1-e^8\right ) x+\left (1+e^{16}+e^{20}\right ) x+x^2-2 x^3+2 e^8 x^3+2 e^{10} x^3+x^5\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 56, normalized size = 2.95 \begin {gather*} x-2 e^8 x+e^{16} x+e^{20} x+2 e^{10} \left (-1+e^8\right ) x+x^2-2 x^3+2 e^8 x^3+2 e^{10} x^3+x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^16 + E^20 + 2*x - 6*x^2 + 5*x^4 + E^8*(-2 + 6*x^2) + E^10*(-2 + 2*E^8 + 6*x^2),x]

[Out]

x - 2*E^8*x + E^16*x + E^20*x + 2*E^10*(-1 + E^8)*x + x^2 - 2*x^3 + 2*E^8*x^3 + 2*E^10*x^3 + x^5

________________________________________________________________________________________

fricas [B]  time = 0.66, size = 48, normalized size = 2.53 \begin {gather*} x^{5} - 2 \, x^{3} + x^{2} + x e^{20} + 2 \, x e^{18} + x e^{16} + 2 \, {\left (x^{3} - x\right )} e^{10} + 2 \, {\left (x^{3} - x\right )} e^{8} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(10)^2+(2*exp(4)^2+6*x^2-2)*exp(10)+exp(4)^4+(6*x^2-2)*exp(4)^2+5*x^4-6*x^2+2*x+1,x, algorithm="f
ricas")

[Out]

x^5 - 2*x^3 + x^2 + x*e^20 + 2*x*e^18 + x*e^16 + 2*(x^3 - x)*e^10 + 2*(x^3 - x)*e^8 + x

________________________________________________________________________________________

giac [B]  time = 0.25, size = 47, normalized size = 2.47 \begin {gather*} x^{5} - 2 \, x^{3} + x^{2} + x e^{20} + x e^{16} + 2 \, {\left (x^{3} + x e^{8} - x\right )} e^{10} + 2 \, {\left (x^{3} - x\right )} e^{8} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(10)^2+(2*exp(4)^2+6*x^2-2)*exp(10)+exp(4)^4+(6*x^2-2)*exp(4)^2+5*x^4-6*x^2+2*x+1,x, algorithm="g
iac")

[Out]

x^5 - 2*x^3 + x^2 + x*e^20 + x*e^16 + 2*(x^3 + x*e^8 - x)*e^10 + 2*(x^3 - x)*e^8 + x

________________________________________________________________________________________

maple [B]  time = 0.06, size = 51, normalized size = 2.68

method	result	size
risch	\(x \,{\mathrm e}^{20}+2 \,{\mathrm e}^{18} x +2 x^{3} {\mathrm e}^{10}-2 x \,{\mathrm e}^{10}+x \,{\mathrm e}^{16}+2 \,{\mathrm e}^{8} x^{3}-2 x \,{\mathrm e}^{8}+x^{5}-2 x^{3}+x^{2}+x\)	\(51\)
norman	\(x^{5}+\left (2 \,{\mathrm e}^{8}+2 \,{\mathrm e}^{10}-2\right ) x^{3}+x^{2}+\left ({\mathrm e}^{16}+2 \,{\mathrm e}^{10} {\mathrm e}^{8}+{\mathrm e}^{20}-2 \,{\mathrm e}^{8}-2 \,{\mathrm e}^{10}+1\right ) x\)	\(54\)
gosper	\(x \left ({\mathrm e}^{16}+2 x^{2} {\mathrm e}^{8}+x^{4}+2 \,{\mathrm e}^{10} {\mathrm e}^{8}+2 \,{\mathrm e}^{10} x^{2}+{\mathrm e}^{20}-2 \,{\mathrm e}^{8}-2 x^{2}-2 \,{\mathrm e}^{10}+x +1\right )\)	\(56\)
default	\(x \,{\mathrm e}^{20}+{\mathrm e}^{10} \left (2 x \,{\mathrm e}^{8}+2 x^{3}-2 x \right )+x \,{\mathrm e}^{16}+{\mathrm e}^{8} \left (2 x^{3}-2 x \right )+x^{5}-2 x^{3}+x^{2}+x\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(10)^2+(2*exp(4)^2+6*x^2-2)*exp(10)+exp(4)^4+(6*x^2-2)*exp(4)^2+5*x^4-6*x^2+2*x+1,x,method=_RETURNVERBO
SE)

[Out]

x*exp(20)+2*exp(18)*x+2*x^3*exp(10)-2*x*exp(10)+x*exp(16)+2*exp(8)*x^3-2*x*exp(8)+x^5-2*x^3+x^2+x

________________________________________________________________________________________

maxima [B]  time = 0.34, size = 47, normalized size = 2.47 \begin {gather*} x^{5} - 2 \, x^{3} + x^{2} + x e^{20} + x e^{16} + 2 \, {\left (x^{3} + x e^{8} - x\right )} e^{10} + 2 \, {\left (x^{3} - x\right )} e^{8} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(10)^2+(2*exp(4)^2+6*x^2-2)*exp(10)+exp(4)^4+(6*x^2-2)*exp(4)^2+5*x^4-6*x^2+2*x+1,x, algorithm="m
axima")

[Out]

x^5 - 2*x^3 + x^2 + x*e^20 + x*e^16 + 2*(x^3 + x*e^8 - x)*e^10 + 2*(x^3 - x)*e^8 + x

________________________________________________________________________________________

mupad [B]  time = 3.43, size = 42, normalized size = 2.21 \begin {gather*} x^5+\left (2\,{\mathrm {e}}^8+2\,{\mathrm {e}}^{10}-2\right )\,x^3+x^2+\left ({\mathrm {e}}^{16}-2\,{\mathrm {e}}^8+{\mathrm {e}}^{20}+{\mathrm {e}}^{10}\,\left (2\,{\mathrm {e}}^8-2\right )+1\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + exp(16) + exp(20) + exp(10)*(2*exp(8) + 6*x^2 - 2) + exp(8)*(6*x^2 - 2) - 6*x^2 + 5*x^4 + 1,x)

[Out]

x^3*(2*exp(8) + 2*exp(10) - 2) + x*(exp(16) - 2*exp(8) + exp(20) + exp(10)*(2*exp(8) - 2) + 1) + x^2 + x^5

________________________________________________________________________________________

sympy [B]  time = 0.06, size = 46, normalized size = 2.42 \begin {gather*} x^{5} + x^{3} \left (-2 + 2 e^{8} + 2 e^{10}\right ) + x^{2} + x \left (- 2 e^{10} - 2 e^{8} + 1 + e^{16} + 2 e^{18} + e^{20}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(10)**2+(2*exp(4)**2+6*x**2-2)*exp(10)+exp(4)**4+(6*x**2-2)*exp(4)**2+5*x**4-6*x**2+2*x+1,x)

[Out]

x**5 + x**3*(-2 + 2*exp(8) + 2*exp(10)) + x**2 + x*(-2*exp(10) - 2*exp(8) + 1 + exp(16) + 2*exp(18) + exp(20))

________________________________________________________________________________________