3.103.31 \(\int \frac {8 e^5 x^{14}+e^{4 x} (16 x^7-8 x^8+e^5 (-14 x^6+8 x^7))}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx\)

Optimal. Leaf size=25 \[ -\frac {2 \left (e^5-x\right )}{\left (4+\frac {e^{4 x}}{x^8}\right ) x} \]

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Rubi [F]  time = 8.82, 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 {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*E^5*x^14 + E^(4*x)*(16*x^7 - 8*x^8 + E^5*(-14*x^6 + 8*x^7)))/(E^(8*x) + 8*E^(4*x)*x^8 + 16*x^16),x]

[Out]

-2*E^5*Defer[Int][(E^x*x^4)/(E^(2*x) - 2*E^x*x^2 + 2*x^4)^2, x] + (2 + E^5)*Defer[Int][(E^x*x^5)/(E^(2*x) - 2*
E^x*x^2 + 2*x^4)^2, x] + 2*E^5*Defer[Int][x^6/(E^(2*x) - 2*E^x*x^2 + 2*x^4)^2, x] - Defer[Int][(E^x*x^6)/(E^(2
*x) - 2*E^x*x^2 + 2*x^4)^2, x] - (2 + E^5)*Defer[Int][x^7/(E^(2*x) - 2*E^x*x^2 + 2*x^4)^2, x] + Defer[Int][x^8
/(E^(2*x) - 2*E^x*x^2 + 2*x^4)^2, x] + (E^5*Defer[Int][E^x/(E^(2*x) - 2*E^x*x^2 + 2*x^4), x])/4 - ((2 + E^5)*D
efer[Int][(E^x*x)/(E^(2*x) - 2*E^x*x^2 + 2*x^4), x])/4 - (E^5*Defer[Int][x^2/(E^(2*x) - 2*E^x*x^2 + 2*x^4), x]
)/2 + Defer[Int][(E^x*x^2)/(E^(2*x) - 2*E^x*x^2 + 2*x^4), x]/4 + ((2 + E^5)*Defer[Int][x^3/(E^(2*x) - 2*E^x*x^
2 + 2*x^4), x])/2 - Defer[Int][x^4/(E^(2*x) - 2*E^x*x^2 + 2*x^4), x]/2 + 2*E^5*Defer[Int][(E^x*x^4)/(E^(2*x) +
2*E^x*x^2 + 2*x^4)^2, x] - (2 + E^5)*Defer[Int][(E^x*x^5)/(E^(2*x) + 2*E^x*x^2 + 2*x^4)^2, x] + 2*E^5*Defer[I
nt][x^6/(E^(2*x) + 2*E^x*x^2 + 2*x^4)^2, x] + Defer[Int][(E^x*x^6)/(E^(2*x) + 2*E^x*x^2 + 2*x^4)^2, x] - (2 +
E^5)*Defer[Int][x^7/(E^(2*x) + 2*E^x*x^2 + 2*x^4)^2, x] + Defer[Int][x^8/(E^(2*x) + 2*E^x*x^2 + 2*x^4)^2, x] -
(E^5*Defer[Int][E^x/(E^(2*x) + 2*E^x*x^2 + 2*x^4), x])/4 + ((2 + E^5)*Defer[Int][(E^x*x)/(E^(2*x) + 2*E^x*x^2
+ 2*x^4), x])/4 - (E^5*Defer[Int][x^2/(E^(2*x) + 2*E^x*x^2 + 2*x^4), x])/2 - Defer[Int][(E^x*x^2)/(E^(2*x) +
2*E^x*x^2 + 2*x^4), x]/4 + ((2 + E^5)*Defer[Int][x^3/(E^(2*x) + 2*E^x*x^2 + 2*x^4), x])/2 - Defer[Int][x^4/(E^
(2*x) + 2*E^x*x^2 + 2*x^4), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{\left (e^{4 x}+4 x^8\right )^2} \, dx\\ &=\int \left (-\frac {\left (e^5-x\right ) (-2+x) x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {\left (e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{4 \left (e^{2 x}-2 e^x x^2+2 x^4\right )}-\frac {\left (e^5-x\right ) (-2+x) x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {\left (-e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{4 \left (e^{2 x}+2 e^x x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{4} \int \frac {\left (-e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\int \frac {\left (e^5-x\right ) (-2+x) x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\int \frac {\left (e^5-x\right ) (-2+x) x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^5 \left (e^x-2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {\left (2+e^5\right ) x \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}-\frac {x^2 \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} \int \left (-\frac {e^5 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {\left (2+e^5\right ) x \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}-\frac {x^2 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx-\int \left (-\frac {2 e^5 x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {\left (2+e^5\right ) x^5 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}-\frac {x^6 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx-\int \left (-\frac {2 e^5 x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {\left (2+e^5\right ) x^5 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}-\frac {x^6 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^2 \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx\right )-\frac {1}{4} \int \frac {x^2 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{4} e^5 \int \frac {e^x-2 x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} e^5 \int \frac {e^x+2 x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\left (2 e^5\right ) \int \frac {x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {x \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {x \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2+e^5\right ) \int \frac {x^5 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {x^5 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^6 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^6 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ &=-\left (\frac {1}{4} \int \left (-\frac {e^x x^2}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {2 x^4}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx\right )-\frac {1}{4} \int \left (\frac {e^x x^2}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^4}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} e^5 \int \left (\frac {e^x}{e^{2 x}-2 e^x x^2+2 x^4}-\frac {2 x^2}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx-\frac {1}{4} e^5 \int \left (\frac {e^x}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^2}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx+\left (2 e^5\right ) \int \left (-\frac {e^x x^4}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\left (2 e^5\right ) \int \left (\frac {e^x x^4}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\frac {1}{4} \left (2+e^5\right ) \int \left (-\frac {e^x x}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {2 x^3}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} \left (2+e^5\right ) \int \left (\frac {e^x x}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^3}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx-\left (2+e^5\right ) \int \left (-\frac {e^x x^5}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^7}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx-\left (2+e^5\right ) \int \left (\frac {e^x x^5}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^7}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\int \left (-\frac {e^x x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^8}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\int \left (\frac {e^x x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^8}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^x x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} \int \frac {e^x x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\frac {1}{2} \int \frac {x^4}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{2} \int \frac {x^4}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{4} e^5 \int \frac {e^x}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} e^5 \int \frac {e^x}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\frac {1}{2} e^5 \int \frac {x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{2} e^5 \int \frac {x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2 e^5\right ) \int \frac {e^x x^4}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {e^x x^4}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (-2-e^5\right ) \int \frac {e^x x}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\left (-2-e^5\right ) \int \frac {e^x x^5}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {e^x x}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2+e^5\right ) \int \frac {x^7}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {e^x x^5}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {x^7}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx-\int \frac {e^x x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {e^x x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.52, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \left (-e^5 x^7+x^8\right )}{e^{4 x}+4 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*E^5*x^14 + E^(4*x)*(16*x^7 - 8*x^8 + E^5*(-14*x^6 + 8*x^7)))/(E^(8*x) + 8*E^(4*x)*x^8 + 16*x^16),
x]

[Out]

(2*(-(E^5*x^7) + x^8))/(E^(4*x) + 4*x^8)

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fricas [A]  time = 0.44, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^8+8*x^8*exp(x)^4+16*x^16),x, a
lgorithm="fricas")

[Out]

2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))

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giac [A]  time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^8+8*x^8*exp(x)^4+16*x^16),x, a
lgorithm="giac")

[Out]

2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))

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maple [A]  time = 0.05, size = 24, normalized size = 0.96

method	result	size
risch	\(-\frac {2 \left ({\mathrm e}^{5}-x \right ) x^{7}}{4 x^{8}+{\mathrm e}^{4 x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^8+8*x^8*exp(x)^4+16*x^16),x,method=_
RETURNVERBOSE)

[Out]

-2*(exp(5)-x)*x^7/(4*x^8+exp(4*x))

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maxima [A]  time = 0.41, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^8+8*x^8*exp(x)^4+16*x^16),x, a
lgorithm="maxima")

[Out]

2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))

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mupad [B]  time = 5.84, size = 50, normalized size = 2.00 \begin {gather*} -\frac {2\,\left (2\,x^{14}\,{\mathrm {e}}^5-x^{15}\,{\mathrm {e}}^5-2\,x^{15}+x^{16}\right )}{\left (2\,x^7-x^8\right )\,\left ({\mathrm {e}}^{4\,x}+4\,x^8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^14*exp(5) - exp(4*x)*(exp(5)*(14*x^6 - 8*x^7) - 16*x^7 + 8*x^8))/(exp(8*x) + 8*x^8*exp(4*x) + 16*x^16
),x)

[Out]

-(2*(2*x^14*exp(5) - x^15*exp(5) - 2*x^15 + x^16))/((2*x^7 - x^8)*(exp(4*x) + 4*x^8))

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sympy [A]  time = 0.12, size = 22, normalized size = 0.88 \begin {gather*} \frac {2 x^{8} - 2 x^{7} e^{5}}{4 x^{8} + e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**7-14*x**6)*exp(5)-8*x**8+16*x**7)*exp(x)**4+8*x**14*exp(5))/(exp(x)**8+8*x**8*exp(x)**4+16*x
**16),x)

[Out]

(2*x**8 - 2*x**7*exp(5))/(4*x**8 + exp(4*x))

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