\(\int e^{\frac {-145-144 x-36 x^2+e^5 (145+144 x+36 x^2)-6 \log (2)}{-1+e^5}} (144+72 x) \, dx\) [1235]

Optimal result

Integrand size = 44, antiderivative size = 28 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=1+e^{1+9 (4+2 x)^2+\frac {6 \log (2)}{1-e^5}} \]

[Out]

1+exp(3*ln(2)/(1/2-1/2*exp(5))+1+3*(4+2*x)*(6*x+12))

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2276, 2268} \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=e^{36 x^2+144 x+\frac {145-145 e^5+\log (64)}{1-e^5}} \]

[In]

Int[E^((-145 - 144*x - 36*x^2 + E^5*(145 + 144*x + 36*x^2) - 6*Log[2])/(-1 + E^5))*(144 + 72*x),x]

[Out]

E^(144*x + 36*x^2 + (145 - 145*E^5 + Log[64])/(1 - E^5))

Rule 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2276

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int e^{144 x+36 x^2+\frac {145-145 e^5+\log (64)}{1-e^5}} (144+72 x) \, dx \\ & = e^{144 x+36 x^2+\frac {145-145 e^5+\log (64)}{1-e^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=2^{-\frac {6}{-1+e^5}} e^{145+144 x+36 x^2} \]

[In]

Integrate[E^((-145 - 144*x - 36*x^2 + E^5*(145 + 144*x + 36*x^2) - 6*Log[2])/(-1 + E^5))*(144 + 72*x),x]

[Out]

E^(145 + 144*x + 36*x^2)/2^(6/(-1 + E^5))

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
risch	\(\left (\frac {1}{64}\right )^{\frac {1}{{\mathrm e}^{5}-1}} {\mathrm e}^{36 x^{2}+144 x +145}\)	\(21\)
derivativedivides	\({\mathrm e}^{\frac {-6 \ln \left (2\right )+\left (36 x^{2}+144 x +145\right ) {\mathrm e}^{5}-36 x^{2}-144 x -145}{{\mathrm e}^{5}-1}}\)	\(36\)
norman	\({\mathrm e}^{\frac {-6 \ln \left (2\right )+\left (36 x^{2}+144 x +145\right ) {\mathrm e}^{5}-36 x^{2}-144 x -145}{{\mathrm e}^{5}-1}}\)	\(36\)
parallelrisch	\({\mathrm e}^{\frac {-6 \ln \left (2\right )+\left (36 x^{2}+144 x +145\right ) {\mathrm e}^{5}-36 x^{2}-144 x -145}{{\mathrm e}^{5}-1}}\)	\(36\)
gosper	\({\mathrm e}^{\frac {36 x^{2} {\mathrm e}^{5}+144 x \,{\mathrm e}^{5}-36 x^{2}+145 \,{\mathrm e}^{5}-6 \ln \left (2\right )-144 x -145}{{\mathrm e}^{5}-1}}\)	\(39\)
parts	\(-\frac {36 i \sqrt {\pi }\, {\mathrm e}^{\frac {-6 \ln \left (2\right )+145 \,{\mathrm e}^{5}-145}{{\mathrm e}^{5}-1}-\frac {\left (144 \,{\mathrm e}^{5}-144\right )^{2}}{4 \left ({\mathrm e}^{5}-1\right ) \left (36 \,{\mathrm e}^{5}-36\right )}} \operatorname {erf}\left (i \sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}\, x +\frac {i \left (144 \,{\mathrm e}^{5}-144\right )}{2 \left ({\mathrm e}^{5}-1\right ) \sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}}\right ) x}{\sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}}-\frac {72 i \sqrt {\pi }\, {\mathrm e}^{\frac {-6 \ln \left (2\right )+145 \,{\mathrm e}^{5}-145}{{\mathrm e}^{5}-1}-\frac {\left (144 \,{\mathrm e}^{5}-144\right )^{2}}{4 \left ({\mathrm e}^{5}-1\right ) \left (36 \,{\mathrm e}^{5}-36\right )}} \operatorname {erf}\left (i \sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}\, x +\frac {i \left (144 \,{\mathrm e}^{5}-144\right )}{2 \left ({\mathrm e}^{5}-1\right ) \sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}}\right )}{\sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}}-\frac {6 i {\mathrm e}^{\frac {-6 \ln \left (2\right )+145 \,{\mathrm e}^{5}-145}{{\mathrm e}^{5}-1}-\frac {\left (144 \,{\mathrm e}^{5}-144\right )^{2}}{4 \left ({\mathrm e}^{5}-1\right ) \left (36 \,{\mathrm e}^{5}-36\right )}} \left (-6 x \,\operatorname {erf}\left (i \left (2+x \right ) \sqrt {36}\right ) \sqrt {\pi }+i {\mathrm e}^{36 \left (2+x \right )^{2}}-12 \,\operatorname {erf}\left (i \left (2+x \right ) \sqrt {36}\right ) \sqrt {\pi }\right )}{\sqrt {\frac {36 \,{\mathrm e}^{5}-36}{{\mathrm e}^{5}-1}}}\)	\(337\)
default	\(72 \,{\mathrm e}^{\frac {145 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}} 2^{-\frac {6}{{\mathrm e}^{5}-1}} {\mathrm e}^{-\frac {145}{{\mathrm e}^{5}-1}} \left (\frac {{\mathrm e}^{\left (\frac {36 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {36}{{\mathrm e}^{5}-1}\right ) x^{2}+\left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right ) x}}{\frac {72 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {72}{{\mathrm e}^{5}-1}}+\frac {i \left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right )^{2}}{4 \left (\frac {36 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {36}{{\mathrm e}^{5}-1}\right )}} \operatorname {erf}\left (6 i \sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}\, x +\frac {i \left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right )}{12 \sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}}\right )}{24 \left (\frac {36 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {36}{{\mathrm e}^{5}-1}\right ) \sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}}\right )-\frac {12 i {\mathrm e}^{\frac {145 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}} 2^{-\frac {6}{{\mathrm e}^{5}-1}} {\mathrm e}^{-\frac {145}{{\mathrm e}^{5}-1}} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right )^{2}}{4 \left (\frac {36 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {36}{{\mathrm e}^{5}-1}\right )}} \operatorname {erf}\left (6 i \sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}\, x +\frac {i \left (\frac {144 \,{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {144}{{\mathrm e}^{5}-1}\right )}{12 \sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}}\right )}{\sqrt {\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}-1}-\frac {1}{{\mathrm e}^{5}-1}}}\)	\(454\)

[In]

int((72*x+144)*exp((-6*ln(2)+(36*x^2+144*x+145)*exp(5)-36*x^2-144*x-145)/(exp(5)-1)),x,method=_RETURNVERBOSE)

[Out]

(1/64)^(1/(exp(5)-1))*exp(36*x^2+144*x+145)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=e^{\left (-\frac {36 \, x^{2} - {\left (36 \, x^{2} + 144 \, x + 145\right )} e^{5} + 144 \, x + 6 \, \log \left (2\right ) + 145}{e^{5} - 1}\right )} \]

[In]

integrate((72*x+144)*exp((-6*log(2)+(36*x^2+144*x+145)*exp(5)-36*x^2-144*x-145)/(exp(5)-1)),x, algorithm="fric
as")

[Out]

e^(-(36*x^2 - (36*x^2 + 144*x + 145)*e^5 + 144*x + 6*log(2) + 145)/(e^5 - 1))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=e^{\frac {- 36 x^{2} - 144 x + \left (36 x^{2} + 144 x + 145\right ) e^{5} - 145 - 6 \log {\left (2 \right )}}{-1 + e^{5}}} \]

[In]

integrate((72*x+144)*exp((-6*ln(2)+(36*x**2+144*x+145)*exp(5)-36*x**2-144*x-145)/(exp(5)-1)),x)

[Out]

exp((-36*x**2 - 144*x + (36*x**2 + 144*x + 145)*exp(5) - 145 - 6*log(2))/(-1 + exp(5)))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.35 (sec) , antiderivative size = 156, normalized size of antiderivative = 5.57 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=-24 i \, \sqrt {\pi } 2^{-\frac {6}{e^{5} - 1} - 1} \operatorname {erf}\left (6 i \, x + 12 i\right ) e^{\left (\frac {145 \, e^{5}}{e^{5} - 1} - \frac {145}{e^{5} - 1} - 144\right )} - {\left (\frac {12 \, \sqrt {\pi } {\left (x + 2\right )} {\left (\operatorname {erf}\left (6 \, \sqrt {-{\left (x + 2\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 2\right )}^{2}}} - e^{\left (36 \, {\left (x + 2\right )}^{2}\right )}\right )} e^{\left (\frac {145 \, e^{5}}{{\left (e^{4} + e^{3} + e^{2} + e + 1\right )} {\left (e - 1\right )}} - \frac {6 \, \log \left (2\right )}{{\left (e^{4} + e^{3} + e^{2} + e + 1\right )} {\left (e - 1\right )}} - \frac {145}{{\left (e^{4} + e^{3} + e^{2} + e + 1\right )} {\left (e - 1\right )}} - 144\right )} \]

[In]

integrate((72*x+144)*exp((-6*log(2)+(36*x^2+144*x+145)*exp(5)-36*x^2-144*x-145)/(exp(5)-1)),x, algorithm="maxi
ma")

[Out]

-24*I*sqrt(pi)*2^(-6/(e^5 - 1) - 1)*erf(6*I*x + 12*I)*e^(145*e^5/(e^5 - 1) - 145/(e^5 - 1) - 144) - (12*sqrt(p
i)*(x + 2)*(erf(6*sqrt(-(x + 2)^2)) - 1)/sqrt(-(x + 2)^2) - e^(36*(x + 2)^2))*e^(145*e^5/((e^4 + e^3 + e^2 + e
+ 1)*(e - 1)) - 6*log(2)/((e^4 + e^3 + e^2 + e + 1)*(e - 1)) - 145/((e^4 + e^3 + e^2 + e + 1)*(e - 1)) - 144)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=e^{\left (\frac {36 \, x^{2} e^{5}}{e^{5} - 1} - \frac {36 \, x^{2}}{e^{5} - 1} + \frac {144 \, x e^{5}}{e^{5} - 1} - \frac {144 \, x}{e^{5} - 1} + \frac {145 \, e^{5}}{e^{5} - 1} - \frac {6 \, \log \left (2\right )}{e^{5} - 1} - \frac {145}{e^{5} - 1}\right )} \]

[In]

integrate((72*x+144)*exp((-6*log(2)+(36*x^2+144*x+145)*exp(5)-36*x^2-144*x-145)/(exp(5)-1)),x, algorithm="giac
")

[Out]

e^(36*x^2*e^5/(e^5 - 1) - 36*x^2/(e^5 - 1) + 144*x*e^5/(e^5 - 1) - 144*x/(e^5 - 1) + 145*e^5/(e^5 - 1) - 6*log
(2)/(e^5 - 1) - 145/(e^5 - 1))

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int e^{\frac {-145-144 x-36 x^2+e^5 \left (145+144 x+36 x^2\right )-6 \log (2)}{-1+e^5}} (144+72 x) \, dx=\frac {{\mathrm {e}}^{\frac {36\,x^2\,{\mathrm {e}}^5}{{\mathrm {e}}^5-1}}\,{\mathrm {e}}^{\frac {145\,{\mathrm {e}}^5}{{\mathrm {e}}^5-1}}\,{\mathrm {e}}^{-\frac {144\,x}{{\mathrm {e}}^5-1}}\,{\mathrm {e}}^{-\frac {36\,x^2}{{\mathrm {e}}^5-1}}\,{\mathrm {e}}^{\frac {144\,x\,{\mathrm {e}}^5}{{\mathrm {e}}^5-1}}\,{\mathrm {e}}^{-\frac {145}{{\mathrm {e}}^5-1}}}{2^{\frac {6}{{\mathrm {e}}^5-1}}} \]

[In]

int(exp(-(144*x + 6*log(2) - exp(5)*(144*x + 36*x^2 + 145) + 36*x^2 + 145)/(exp(5) - 1))*(72*x + 144),x)

[Out]

(exp((36*x^2*exp(5))/(exp(5) - 1))*exp((145*exp(5))/(exp(5) - 1))*exp(-(144*x)/(exp(5) - 1))*exp(-(36*x^2)/(ex
p(5) - 1))*exp((144*x*exp(5))/(exp(5) - 1))*exp(-145/(exp(5) - 1)))/2^(6/(exp(5) - 1))