3.14.13 \(\int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} (1+2 e^{x^2} x+e^{36 x^4} (288 x^3-144 e^{x^2} x^3-144 x^4))}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx\)

Optimal. Leaf size=24 \[ \frac {-2+e^{x^2}+x}{5+e^{5+e^{36 x^4}}} \]

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Rubi [F]  time = 3.67, 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 {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 + 10*E^x^2*x + E^(5 + E^(36*x^4))*(1 + 2*E^x^2*x + E^(36*x^4)*(288*x^3 - 144*E^x^2*x^3 - 144*x^4)))/(25
+ 10*E^(5 + E^(36*x^4)) + E^(10 + 2*E^(36*x^4))),x]

[Out]

-2/(5 + E^(5 + E^(36*x^4))) + Defer[Int][(5 + E^(5 + E^(36*x^4)))^(-1), x] - 144*Defer[Int][(E^(5 + E^(36*x^4)
+ 36*x^4)*x^4)/(5 + E^(5 + E^(36*x^4)))^2, x] + 5*Defer[Subst][Defer[Int][E^x/(5 + E^(5 + E^(36*x^2)))^2, x],
x, x^2] + Defer[Subst][Defer[Int][E^(5 + E^(36*x^2) + x)/(5 + E^(5 + E^(36*x^2)))^2, x], x, x^2] - 72*Defer[S
ubst][Defer[Int][(E^(5 + E^(36*x^2) + x + 36*x^2)*x)/(5 + E^(5 + E^(36*x^2)))^2, x], x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x-144 e^{36 x^4} x^3 \left (-2+e^{x^2}+x\right )\right )}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx\\ &=\int \left (\frac {5}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {e^{5+e^{36 x^4}}}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {10 e^{x^2} x}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {2 e^{5+e^{36 x^4}+x^2} x}{\left (5+e^{5+e^{36 x^4}}\right )^2}-\frac {144 e^{5+e^{36 x^4}+36 x^4} x^3 \left (-2+e^{x^2}+x\right )}{\left (5+e^{5+e^{36 x^4}}\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{5+e^{36 x^4}+x^2} x}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+5 \int \frac {1}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+10 \int \frac {e^{x^2} x}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx-144 \int \frac {e^{5+e^{36 x^4}+36 x^4} x^3 \left (-2+e^{x^2}+x\right )}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+\int \frac {e^{5+e^{36 x^4}}}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx\\ &=5 \int \frac {1}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+5 \operatorname {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-144 \int \left (\frac {e^{5+e^{36 x^4}+x^2+36 x^4} x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {e^{5+e^{36 x^4}+36 x^4} (-2+x) x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2}\right ) \, dx+\int \left (-\frac {5}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {1}{5+e^{5+e^{36 x^4}}}\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x}}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )\\ &=5 \operatorname {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-144 \int \frac {e^{5+e^{36 x^4}+x^2+36 x^4} x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx-144 \int \frac {e^{5+e^{36 x^4}+36 x^4} (-2+x) x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+\int \frac {1}{5+e^{5+e^{36 x^4}}} \, dx+\operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x}}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )\\ &=5 \operatorname {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-72 \operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x+36 x^2} x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-144 \int \left (-\frac {2 e^{5+e^{36 x^4}+36 x^4} x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2}+\frac {e^{5+e^{36 x^4}+36 x^4} x^4}{\left (5+e^{5+e^{36 x^4}}\right )^2}\right ) \, dx+\int \frac {1}{5+e^{5+e^{36 x^4}}} \, dx+\operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x}}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )\\ &=5 \operatorname {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-72 \operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x+36 x^2} x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-144 \int \frac {e^{5+e^{36 x^4}+36 x^4} x^4}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+288 \int \frac {e^{5+e^{36 x^4}+36 x^4} x^3}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+\int \frac {1}{5+e^{5+e^{36 x^4}}} \, dx+\operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x}}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )\\ &=-\frac {2}{5+e^{5+e^{36 x^4}}}+5 \operatorname {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-72 \operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x+36 x^2} x}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )-144 \int \frac {e^{5+e^{36 x^4}+36 x^4} x^4}{\left (5+e^{5+e^{36 x^4}}\right )^2} \, dx+\int \frac {1}{5+e^{5+e^{36 x^4}}} \, dx+\operatorname {Subst}\left (\int \frac {e^{5+e^{36 x^2}+x}}{\left (5+e^{5+e^{36 x^2}}\right )^2} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.71, size = 24, normalized size = 1.00 \begin {gather*} \frac {-2+e^{x^2}+x}{5+e^{5+e^{36 x^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 10*E^x^2*x + E^(5 + E^(36*x^4))*(1 + 2*E^x^2*x + E^(36*x^4)*(288*x^3 - 144*E^x^2*x^3 - 144*x^4)
))/(25 + 10*E^(5 + E^(36*x^4)) + E^(10 + 2*E^(36*x^4))),x]

[Out]

(-2 + E^x^2 + x)/(5 + E^(5 + E^(36*x^4)))

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fricas [A]  time = 0.70, size = 21, normalized size = 0.88 \begin {gather*} \frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5
)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5)+25),x, algorithm="fricas")

[Out]

(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)

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giac [A]  time = 0.33, size = 21, normalized size = 0.88 \begin {gather*} \frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5
)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5)+25),x, algorithm="giac")

[Out]

(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)

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maple [A]  time = 0.07, size = 22, normalized size = 0.92

method	result	size
risch	\(\frac {x -2+{\mathrm e}^{x^{2}}}{5+{\mathrm e}^{{\mathrm e}^{36 x^{4}}+5}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp
(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5)+25),x,method=_RETURNVERBOSE)

[Out]

(x-2+exp(x^2))/(5+exp(exp(36*x^4)+5))

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maxima [A]  time = 0.46, size = 21, normalized size = 0.88 \begin {gather*} \frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5
)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5)+25),x, algorithm="maxima")

[Out]

(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)

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mupad [B]  time = 1.05, size = 59, normalized size = 2.46 \begin {gather*} \frac {{\mathrm {e}}^{-36\,x^4}\,\left (x^3\,{\mathrm {e}}^{36\,x^4+x^2}-2\,x^3\,{\mathrm {e}}^{36\,x^4}+x^4\,{\mathrm {e}}^{36\,x^4}\right )}{x^3\,\left ({\mathrm {e}}^{{\mathrm {e}}^{36\,x^4}+5}+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x*exp(x^2) + exp(exp(36*x^4) + 5)*(2*x*exp(x^2) - exp(36*x^4)*(144*x^3*exp(x^2) - 288*x^3 + 144*x^4) +
1) + 5)/(10*exp(exp(36*x^4) + 5) + exp(2*exp(36*x^4) + 10) + 25),x)

[Out]

(exp(-36*x^4)*(x^3*exp(x^2 + 36*x^4) - 2*x^3*exp(36*x^4) + x^4*exp(36*x^4)))/(x^3*(exp(exp(36*x^4) + 5) + 5))

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sympy [A]  time = 0.31, size = 19, normalized size = 0.79 \begin {gather*} \frac {x + e^{x^{2}} - 2}{e^{e^{36 x^{4}} + 5} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-144*x**3*exp(x**2)-144*x**4+288*x**3)*exp(36*x**4)+2*exp(x**2)*x+1)*exp(exp(36*x**4)+5)+10*exp(x
**2)*x+5)/(exp(exp(36*x**4)+5)**2+10*exp(exp(36*x**4)+5)+25),x)

[Out]

(x + exp(x**2) - 2)/(exp(exp(36*x**4) + 5) + 5)

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