3.67.30 \(\int \frac {e^{e^{3+x}} (8 \log ^7(4+x)+e^{3+x} (4+x) \log ^8(4+x))}{4+x} \, dx\)

Optimal. Leaf size=16 \[ 6+e^{e^{3+x}} \log ^8(4+x) \]

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Rubi [A]  time = 0.12, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2288} \begin {gather*} e^{e^{x+3}} \log ^8(x+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^(3 + x)*(8*Log[4 + x]^7 + E^(3 + x)*(4 + x)*Log[4 + x]^8))/(4 + x),x]

[Out]

E^E^(3 + x)*Log[4 + x]^8

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} &=e^{e^{3+x}} \log ^8(4+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 14, normalized size = 0.88 \begin {gather*} e^{e^{3+x}} \log ^8(4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(3 + x)*(8*Log[4 + x]^7 + E^(3 + x)*(4 + x)*Log[4 + x]^8))/(4 + x),x]

[Out]

E^E^(3 + x)*Log[4 + x]^8

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fricas [A]  time = 0.56, size = 12, normalized size = 0.75 \begin {gather*} e^{\left (e^{\left (x + 3\right )}\right )} \log \left (x + 4\right )^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+x)*exp(3+x)*log(4+x)^8+8*log(4+x)^7)*exp(exp(3+x))/(4+x),x, algorithm="fricas")

[Out]

e^(e^(x + 3))*log(x + 4)^8

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giac [A]  time = 0.22, size = 12, normalized size = 0.75 \begin {gather*} e^{\left (e^{\left (x + 3\right )}\right )} \log \left (x + 4\right )^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+x)*exp(3+x)*log(4+x)^8+8*log(4+x)^7)*exp(exp(3+x))/(4+x),x, algorithm="giac")

[Out]

e^(e^(x + 3))*log(x + 4)^8

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maple [A]  time = 0.07, size = 13, normalized size = 0.81

method	result	size
risch	\(\ln \left (4+x \right )^{8} {\mathrm e}^{{\mathrm e}^{3+x}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4+x)*exp(3+x)*ln(4+x)^8+8*ln(4+x)^7)*exp(exp(3+x))/(4+x),x,method=_RETURNVERBOSE)

[Out]

ln(4+x)^8*exp(exp(3+x))

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maxima [A]  time = 0.43, size = 12, normalized size = 0.75 \begin {gather*} e^{\left (e^{\left (x + 3\right )}\right )} \log \left (x + 4\right )^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+x)*exp(3+x)*log(4+x)^8+8*log(4+x)^7)*exp(exp(3+x))/(4+x),x, algorithm="maxima")

[Out]

e^(e^(x + 3))*log(x + 4)^8

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {{\mathrm {e}}^{{\mathrm {e}}^{x+3}}\,\left ({\mathrm {e}}^{x+3}\,\left (x+4\right )\,{\ln \left (x+4\right )}^8+8\,{\ln \left (x+4\right )}^7\right )}{x+4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x + 3))*(8*log(x + 4)^7 + log(x + 4)^8*exp(x + 3)*(x + 4)))/(x + 4),x)

[Out]

int((exp(exp(x + 3))*(8*log(x + 4)^7 + log(x + 4)^8*exp(x + 3)*(x + 4)))/(x + 4), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+x)*exp(3+x)*ln(4+x)**8+8*ln(4+x)**7)*exp(exp(3+x))/(4+x),x)

[Out]

Timed out

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