3.7.1 \(\int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} (-36 x^2+24 x^3-3 x^4)+e^x (36 x^2-24 x^3+3 x^4)}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} (10-2 e^x)} \, dx\) [601]

3.7.1.1 Optimal result

Integrand size = 104, antiderivative size = 24 \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\frac {3 (-4+x) x^3}{5-e^x+e^{\frac {3}{4}+x}} \]

output

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

3.7.1.2 Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\frac {3 (-4+x) x^3}{5-e^x+e^{\frac {3}{4}+x}} \]

input

Integrate[(-180*x^2 + 60*x^3 + E^((3 + 4*x)/4)*(-36*x^2 + 24*x^3 - 3*x^4) 
+ E^x*(36*x^2 - 24*x^3 + 3*x^4))/(25 - 10*E^x + E^(2*x) + E^((3 + 4*x)/2) 
+ E^((3 + 4*x)/4)*(10 - 2*E^x)),x]
 

output

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

3.7.1.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).

Time = 2.70 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {7239, 27, 25, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {60 x^3-180 x^2+e^{\frac {1}{4} (4 x+3)} \left (-3 x^4+24 x^3-36 x^2\right )+e^x \left (3 x^4-24 x^3+36 x^2\right )}{e^{\frac {1}{4} (4 x+3)} \left (10-2 e^x\right )-10 e^x+e^{2 x}+e^{\frac {1}{2} (4 x+3)}+25} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {3 x^2 \left (\left (1-e^{3/4}\right ) e^x \left (x^2-8 x+12\right )+20 (x-3)\right )}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int -\frac {x^2 \left (20 (3-x)-\left (1-e^{3/4}\right ) e^x \left (x^2-8 x+12\right )\right )}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -3 \int \frac {x^2 \left (20 (3-x)-\left (1-e^{3/4}\right ) e^x \left (x^2-8 x+12\right )\right )}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {5 (4-x) x^3}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}+\frac {\left (x^2-8 x+12\right ) x^2}{5-\left (1-e^{3/4}\right ) e^x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 \left (\frac {4 x^3}{5-\left (1-e^{3/4}\right ) e^x}-\frac {x^4}{5-\left (1-e^{3/4}\right ) e^x}\right )\)

input

Int[(-180*x^2 + 60*x^3 + E^((3 + 4*x)/4)*(-36*x^2 + 24*x^3 - 3*x^4) + E^x* 
(36*x^2 - 24*x^3 + 3*x^4))/(25 - 10*E^x + E^(2*x) + E^((3 + 4*x)/2) + E^(( 
3 + 4*x)/4)*(10 - 2*E^x)),x]
 

output

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

3.7.1.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.7.1.4 Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {3 x^{3} \left (x -4\right )}{5-{\mathrm e}^{x}+{\mathrm e}^{\frac {3}{4}+x}}\)	\(21\)
norman	\(\frac {3 x^{4}-12 x^{3}}{{\mathrm e}^{\frac {3}{4}} {\mathrm e}^{x}-{\mathrm e}^{x}+5}\)	\(26\)
parallelrisch	\(-\frac {3 x^{4}-12 x^{3}}{-5+{\mathrm e}^{x}-{\mathrm e}^{\frac {3}{4}+x}}\)	\(26\)

input

int(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+60*x^3 
-180*x^2)/(exp(3/4+x)^2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x)+25),x 
,method=_RETURNVERBOSE)
 

output

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

3.7.1.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\frac {3 \, {\left (x^{4} - 4 \, x^{3}\right )} e^{\frac {3}{4}}}{{\left (e^{\frac {3}{4}} - 1\right )} e^{\left (x + \frac {3}{4}\right )} + 5 \, e^{\frac {3}{4}}} \]

input

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+ 
60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x) 
+25),x, algorithm=\
 

output

3*(x^4 - 4*x^3)*e^(3/4)/((e^(3/4) - 1)*e^(x + 3/4) + 5*e^(3/4))
 

3.7.1.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\text {Exception raised: PolynomialError} \]

input

integrate(((-3*x**4+24*x**3-36*x**2)*exp(3/4+x)+(3*x**4-24*x**3+36*x**2)*e 
xp(x)+60*x**3-180*x**2)/(exp(3/4+x)**2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)**2 
-10*exp(x)+25),x)
 

output

Exception raised: PolynomialError >> (-3*_t0*x**4*exp(3/4) + 3*_t0*x**4 - 
24*_t0*x**3 + 24*_t0*x**3*exp(3/4) - 36*_t0*x**2*exp(3/4) + 36*_t0*x**2 + 
60*x**3 - 180*x**2)/(-2*_t0**2*exp(3/4) + _t0**2 + _t0**2*exp(3/2) - 10*_t 
0 + 10*_t0*exp(
 

3.7.1.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+ 
60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x) 
+25),x, algorithm=\
 

output

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

3.7.1.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\frac {3 \, {\left ({\left (4 \, x + 3\right )}^{4} - 28 \, {\left (4 \, x + 3\right )}^{3} + 198 \, {\left (4 \, x + 3\right )}^{2} - 2160 \, x - 1107\right )}}{256 \, {\left (e^{\left (x + \frac {3}{4}\right )} - e^{x} + 5\right )}} \]

input

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+ 
60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x) 
+25),x, algorithm=\
 

output

3/256*((4*x + 3)^4 - 28*(4*x + 3)^3 + 198*(4*x + 3)^2 - 2160*x - 1107)/(e^ 
(x + 3/4) - e^x + 5)
 

3.7.1.9 Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} \left (-36 x^2+24 x^3-3 x^4\right )+e^x \left (36 x^2-24 x^3+3 x^4\right )}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} \left (10-2 e^x\right )} \, dx=\frac {3\,x^3\,\left (x-4\right )}{{\mathrm {e}}^x\,\left ({\mathrm {e}}^{3/4}-1\right )+5} \]

input

int((exp(x)*(36*x^2 - 24*x^3 + 3*x^4) - exp(x + 3/4)*(36*x^2 - 24*x^3 + 3* 
x^4) - 180*x^2 + 60*x^3)/(exp(2*x) + exp(2*x + 3/2) - 10*exp(x) - exp(x + 
3/4)*(2*exp(x) - 10) + 25),x)
 

output

(3*x^3*(x - 4))/(exp(x)*(exp(3/4) - 1) + 5)