3.69.53 $\int \frac{-6x-16x^3+e^{10}(1-6x+4x^2-16x^3)+e^5(12x+32x^3)}{1-2e^5+e^{10}}dx$

Optimal. Leaf size=26 $$(x^2+\frac{4x^4}{3})(-3+\frac{x}{{(x-\frac{x}{e^5})}^2})$$

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Rubi [B]  time = 0.03, antiderivative size = 130, normalized size of antiderivative = 5.00, number of steps used = 4, number of rules used = 1, integrand size = 54, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.019, Rules used = {12} $$\begin{array}{c}-\frac{4e^{10}{x}^4}{{(1-e^5)}^2}+\frac{8e^5{x}^4}{{(1-e^5)}^2}-\frac{4x^4}{{(1-e^5)}^2}+\frac{4e^{10}{x}^3}{3{(1-e^5)}^2}-\frac{3e^{10}{x}^2}{{(1-e^5)}^2}+\frac{6e^5{x}^2}{{(1-e^5)}^2}-\frac{3x^2}{{(1-e^5)}^2}+\frac{e^{10}x}{{(1-e^5)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-6*x - 16*x^3 + E^10*(1 - 6*x + 4*x^2 - 16*x^3) + E^5*(12*x + 32*x^3))/(1 - 2*E^5 + E^10),x]

[Out]

(E^10*x)/(1 - E^5)^2 - (3*x^2)/(1 - E^5)^2 + (6*E^5*x^2)/(1 - E^5)^2 - (3*E^10*x^2)/(1 - E^5)^2 + (4*E^10*x^3)
/(3*(1 - E^5)^2) - (4*x^4)/(1 - E^5)^2 + (8*E^5*x^4)/(1 - E^5)^2 - (4*E^10*x^4)/(1 - E^5)^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int (-6x-16x^3+e^{10}(1-6x+4x^2-16x^3)+e^5(12x+32x^3))dx}{1-2e^5+e^{10}}\\ & =-\frac{3x^2}{{(1-e^5)}^2}-\frac{4x^4}{{(1-e^5)}^2}+\frac{e^5\int (12x+32x^3)dx}{{(1-e^5)}^2}+\frac{e^{10}\int (1-6x+4x^2-16x^3)dx}{{(1-e^5)}^2}\\ & =\frac{e^{10}x}{{(1-e^5)}^2}-\frac{3x^2}{{(1-e^5)}^2}+\frac{6e^5{x}^2}{{(1-e^5)}^2}-\frac{3e^{10}{x}^2}{{(1-e^5)}^2}+\frac{4e^{10}{x}^3}{3{(1-e^5)}^2}-\frac{4x^4}{{(1-e^5)}^2}+\frac{8e^5{x}^4}{{(1-e^5)}^2}-\frac{4e^{10}{x}^4}{{(1-e^5)}^2}\end{array}\end{array}$$

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Mathematica [B]  time = 0.01, size = 66, normalized size = 2.54 $$\begin{array}{c}\frac{e^{10}x-3x^2+6e^5{x}^2-3e^{10}{x}^2+\frac{4e^{10}{x}^3}{3}-4x^4+8e^5{x}^4-4e^{10}{x}^4}{{(-1+e^5)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-6*x - 16*x^3 + E^10*(1 - 6*x + 4*x^2 - 16*x^3) + E^5*(12*x + 32*x^3))/(1 - 2*E^5 + E^10),x]

[Out]

(E^10*x - 3*x^2 + 6*E^5*x^2 - 3*E^10*x^2 + (4*E^10*x^3)/3 - 4*x^4 + 8*E^5*x^4 - 4*E^10*x^4)/(-1 + E^5)^2

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fricas [B]  time = 0.60, size = 60, normalized size = 2.31 $$\begin{array}{c}-\frac{12x^4+9x^2+(12x^4-4x^3+9x^2-3x)e^{10}-6(4x^4+3x^2)e^5}{3(e^{10}-2e^5+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3+4*x^2-6*x+1)*exp(5)^2+(32*x^3+12*x)*exp(5)-16*x^3-6*x)/(exp(5)^2-2*exp(5)+1),x, algorithm=
"fricas")

[Out]

-1/3*(12*x^4 + 9*x^2 + (12*x^4 - 4*x^3 + 9*x^2 - 3*x)*e^10 - 6*(4*x^4 + 3*x^2)*e^5)/(e^10 - 2*e^5 + 1)

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giac [B]  time = 0.15, size = 60, normalized size = 2.31 $$\begin{array}{c}-\frac{12x^4+9x^2+(12x^4-4x^3+9x^2-3x)e^{10}-6(4x^4+3x^2)e^5}{3(e^{10}-2e^5+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3+4*x^2-6*x+1)*exp(5)^2+(32*x^3+12*x)*exp(5)-16*x^3-6*x)/(exp(5)^2-2*exp(5)+1),x, algorithm=
"giac")

[Out]

-1/3*(12*x^4 + 9*x^2 + (12*x^4 - 4*x^3 + 9*x^2 - 3*x)*e^10 - 6*(4*x^4 + 3*x^2)*e^5)/(e^10 - 2*e^5 + 1)

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maple [B]  time = 0.04, size = 56, normalized size = 2.15

method	result	size
norman	$\frac{(-4{\mathrm{e}}^5+4)x^4+(-3{\mathrm{e}}^5+3)x^2+\frac{{\mathrm{e}}^{10}x}{{\mathrm{e}}^5-1}+\frac{4{\mathrm{e}}^{10}{x}^3}{3({\mathrm{e}}^5-1)}}{{\mathrm{e}}^5-1}$	$56$
default	$\frac{{\mathrm{e}}^{10}(-4x^4+\frac{4}{3}{x}^3-3x^2+x)+{\mathrm{e}}^5(8x^4+6x^2)-4x^4-3x^2}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}$	$61$
gosper	$-\frac{x(12x^3{\mathrm{e}}^{10}-4x^2{\mathrm{e}}^{10}-24x^3{\mathrm{e}}^5+9x{\mathrm{e}}^{10}+12x^3-3{\mathrm{e}}^{10}-18x{\mathrm{e}}^5+9x)}{3({\mathrm{e}}^{10}-2{\mathrm{e}}^5+1)}$	$68$
risch	$-\frac{4x^4{\mathrm{e}}^{10}}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}+\frac{4x^3{\mathrm{e}}^{10}}{3({\mathrm{e}}^{10}-2{\mathrm{e}}^5+1)}+\frac{8x^4{\mathrm{e}}^5}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}-\frac{3x^2{\mathrm{e}}^{10}}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}-\frac{4x^4}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}+\frac{x{\mathrm{e}}^{10}}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}+\frac{6x^2{\mathrm{e}}^5}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}-\frac{3x^2}{{\mathrm{e}}^{10}-2{\mathrm{e}}^5+1}$	$131$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^3+4*x^2-6*x+1)*exp(5)^2+(32*x^3+12*x)*exp(5)-16*x^3-6*x)/(exp(5)^2-2*exp(5)+1),x,method=_RETURNVER
BOSE)

[Out]

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

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maxima [B]  time = 0.36, size = 60, normalized size = 2.31 $$\begin{array}{c}-\frac{12x^4+9x^2+(12x^4-4x^3+9x^2-3x)e^{10}-6(4x^4+3x^2)e^5}{3(e^{10}-2e^5+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3+4*x^2-6*x+1)*exp(5)^2+(32*x^3+12*x)*exp(5)-16*x^3-6*x)/(exp(5)^2-2*exp(5)+1),x, algorithm=
"maxima")

[Out]

-1/3*(12*x^4 + 9*x^2 + (12*x^4 - 4*x^3 + 9*x^2 - 3*x)*e^10 - 6*(4*x^4 + 3*x^2)*e^5)/(e^10 - 2*e^5 + 1)

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mupad [B]  time = 4.10, size = 34, normalized size = 1.31 $$\begin{array}{c}-4x^4+\frac{4{\mathrm{e}}^{10}{x}^3}{3{({\mathrm{e}}^5-1)}^2}-3x^2+\frac{{\mathrm{e}}^{10}x}{{({\mathrm{e}}^5-1)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - exp(5)*(12*x + 32*x^3) + exp(10)*(6*x - 4*x^2 + 16*x^3 - 1) + 16*x^3)/(exp(10) - 2*exp(5) + 1),x)

[Out]

(x*exp(10))/(exp(5) - 1)^2 - 4*x^4 - 3*x^2 + (4*x^3*exp(10))/(3*(exp(5) - 1)^2)

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sympy [B]  time = 0.08, size = 44, normalized size = 1.69 $$\begin{array}{c}-4x^4+\frac{4x^3{e}^{10}}{-6e^5+3+3e^{10}}-3x^2+\frac{x{e}^{10}}{-2e^5+1+e^{10}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**3+4*x**2-6*x+1)*exp(5)**2+(32*x**3+12*x)*exp(5)-16*x**3-6*x)/(exp(5)**2-2*exp(5)+1),x)

[Out]

-4*x**4 + 4*x**3*exp(10)/(-6*exp(5) + 3 + 3*exp(10)) - 3*x**2 + x*exp(10)/(-2*exp(5) + 1 + exp(10))

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