3.78.13 \(\int \frac {-1+e^{2+e^x x^4} (2 x^3+e^x (-8 x^5-2 x^6+4 x^7+x^8))}{x^2} \, dx\)

Optimal. Leaf size=22 \[ -2+\frac {1}{x}+e^{2+e^x x^4} \left (-2+x^2\right ) \]

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Rubi [B]  time = 0.31, antiderivative size = 69, normalized size of antiderivative = 3.14, number of steps used = 3, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {14, 2288} \begin {gather*} \frac {1}{x}-\frac {e^{e^x x^4+2} x \left (-e^x x^5-4 e^x x^4+2 e^x x^3+8 e^x x^2\right )}{e^x x^4+4 e^x x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^(2 + E^x*x^4)*(2*x^3 + E^x*(-8*x^5 - 2*x^6 + 4*x^7 + x^8)))/x^2,x]

[Out]

x^(-1) - (E^(2 + E^x*x^4)*x*(8*E^x*x^2 + 2*E^x*x^3 - 4*E^x*x^4 - E^x*x^5))/(4*E^x*x^3 + E^x*x^4)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x^2}+e^{2+e^x x^4} x \left (2-8 e^x x^2-2 e^x x^3+4 e^x x^4+e^x x^5\right )\right ) \, dx\\ &=\frac {1}{x}+\int e^{2+e^x x^4} x \left (2-8 e^x x^2-2 e^x x^3+4 e^x x^4+e^x x^5\right ) \, dx\\ &=\frac {1}{x}-\frac {e^{2+e^x x^4} x \left (8 e^x x^2+2 e^x x^3-4 e^x x^4-e^x x^5\right )}{4 e^x x^3+e^x x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{x}+e^{2+e^x x^4} \left (-2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^(2 + E^x*x^4)*(2*x^3 + E^x*(-8*x^5 - 2*x^6 + 4*x^7 + x^8)))/x^2,x]

[Out]

x^(-1) + E^(2 + E^x*x^4)*(-2 + x^2)

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fricas [A]  time = 0.89, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (x^{3} - 2 \, x\right )} e^{\left (x^{4} e^{x} + 2\right )} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^8+4*x^7-2*x^6-8*x^5)*exp(x)+2*x^3)*exp(exp(x)*x^4+2)-1)/x^2,x, algorithm="fricas")

[Out]

((x^3 - 2*x)*e^(x^4*e^x + 2) + 1)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} + {\left (x^{8} + 4 \, x^{7} - 2 \, x^{6} - 8 \, x^{5}\right )} e^{x}\right )} e^{\left (x^{4} e^{x} + 2\right )} - 1}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^8+4*x^7-2*x^6-8*x^5)*exp(x)+2*x^3)*exp(exp(x)*x^4+2)-1)/x^2,x, algorithm="giac")

[Out]

integrate(((2*x^3 + (x^8 + 4*x^7 - 2*x^6 - 8*x^5)*e^x)*e^(x^4*e^x + 2) - 1)/x^2, x)

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maple [A]  time = 0.05, size = 20, normalized size = 0.91

method	result	size
risch	\(\frac {1}{x}+\left (x^{2}-2\right ) {\mathrm e}^{{\mathrm e}^{x} x^{4}+2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^8+4*x^7-2*x^6-8*x^5)*exp(x)+2*x^3)*exp(exp(x)*x^4+2)-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 23, normalized size = 1.05 \begin {gather*} {\left (x^{2} e^{2} - 2 \, e^{2}\right )} e^{\left (x^{4} e^{x}\right )} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^8+4*x^7-2*x^6-8*x^5)*exp(x)+2*x^3)*exp(exp(x)*x^4+2)-1)/x^2,x, algorithm="maxima")

[Out]

(x^2*e^2 - 2*e^2)*e^(x^4*e^x) + 1/x

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mupad [B]  time = 5.21, size = 19, normalized size = 0.86 \begin {gather*} {\mathrm {e}}^{x^4\,{\mathrm {e}}^x+2}\,\left (x^2-2\right )+\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^4*exp(x) + 2)*(exp(x)*(8*x^5 + 2*x^6 - 4*x^7 - x^8) - 2*x^3) + 1)/x^2,x)

[Out]

exp(x^4*exp(x) + 2)*(x^2 - 2) + 1/x

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sympy [A]  time = 0.23, size = 17, normalized size = 0.77 \begin {gather*} \left (x^{2} - 2\right ) e^{x^{4} e^{x} + 2} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**8+4*x**7-2*x**6-8*x**5)*exp(x)+2*x**3)*exp(exp(x)*x**4+2)-1)/x**2,x)

[Out]

(x**2 - 2)*exp(x**4*exp(x) + 2) + 1/x

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