3.98.4 \(\int \frac {(9+e-x^2-18 x^4+8 x^5-7 x^8) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \]

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Rubi [F]  time = 1.26, 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 {\left (9+e-x^2-18 x^4+8 x^5-7 x^8\right ) (i \pi +\log (3))}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((9 + E - x^2 - 18*x^4 + 8*x^5 - 7*x^8)*(I*Pi + Log[3]))/(81 + E^2 - 108*x + 54*x^2 - 12*x^3 + 109*x^4 - 1
08*x^5 + 36*x^6 - 4*x^7 + 54*x^8 - 36*x^9 + 6*x^10 + 12*x^12 - 4*x^13 + x^16 + E*(18 - 12*x + 2*x^2 + 12*x^4 -
4*x^5 + 2*x^8)),x]

[Out]

(-2*(9 + E)*(I*Pi + Log[3])*Defer[Int][(-3 + I*Sqrt[E] + x - x^4)^(-2), x])/E - (7*(I*Pi + Log[3])*Defer[Int][
(-3*I + Sqrt[E] + I*x - I*x^4)^(-1), x])/(2*Sqrt[E]) + (2*(9 + E)*(I*Pi + Log[3])*Defer[Int][(-3*I + Sqrt[E] +
I*x - I*x^4)^(-1), x])/E^(3/2) - (7*(I*Pi + Log[3])*Defer[Int][(3*I + Sqrt[E] - I*x + I*x^4)^(-1), x])/(2*Sqr
t[E]) + (2*(9 + E)*(I*Pi + Log[3])*Defer[Int][(3*I + Sqrt[E] - I*x + I*x^4)^(-1), x])/E^(3/2) - (2*(9 + E)*(I*
Pi + Log[3])*Defer[Int][(3 + I*Sqrt[E] - x + x^4)^(-2), x])/E - 42*(I*Pi + Log[3])*Defer[Int][x/(E + (3 - x +
x^4)^2)^2, x] + 6*(I*Pi + Log[3])*Defer[Int][x^2/(E + (3 - x + x^4)^2)^2, x] + 24*(I*Pi + Log[3])*Defer[Int][x
^4/(E + (3 - x + x^4)^2)^2, x] - 6*(I*Pi + Log[3])*Defer[Int][x^5/(E + (3 - x + x^4)^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=(i \pi +\log (3)) \int \frac {9+e-x^2-18 x^4+8 x^5-7 x^8}{81+e^2-108 x+54 x^2-12 x^3+109 x^4-108 x^5+36 x^6-4 x^7+54 x^8-36 x^9+6 x^{10}+12 x^{12}-4 x^{13}+x^{16}+e \left (18-12 x+2 x^2+12 x^4-4 x^5+2 x^8\right )} \, dx\\ &=(i \pi +\log (3)) \int \left (\frac {7}{-9 \left (1+\frac {e}{9}\right )+6 x-x^2-6 x^4+2 x^5-x^8}+\frac {2 \left (36 \left (1+\frac {e}{9}\right )-21 x+3 x^2+12 x^4-3 x^5\right )}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}\right ) \, dx\\ &=(2 (i \pi +\log (3))) \int \frac {36 \left (1+\frac {e}{9}\right )-21 x+3 x^2+12 x^4-3 x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx+(7 (i \pi +\log (3))) \int \frac {1}{-9 \left (1+\frac {e}{9}\right )+6 x-x^2-6 x^4+2 x^5-x^8} \, dx\\ &=(2 (i \pi +\log (3))) \int \frac {4 e+3 \left (12-7 x+x^2+4 x^4-x^5\right )}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(7 (i \pi +\log (3))) \int \frac {1}{-e-\left (3-x+x^4\right )^2} \, dx\\ &=(2 (i \pi +\log (3))) \int \left (\frac {36 \left (1+\frac {e}{9}\right )}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}-\frac {21 x}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}+\frac {3 x^2}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}+\frac {12 x^4}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}-\frac {3 x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2}\right ) \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {\sqrt {e}}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 e}-\frac {(7 (i \pi +\log (3))) \int \frac {\sqrt {e}}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 e}\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (9 \left (1+\frac {e}{9}\right )-6 x+x^2+6 x^4-2 x^5+x^8\right )^2} \, dx\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+(8 (9+e) (i \pi +\log (3))) \int \left (-\frac {1}{4 e \left (-3+i \sqrt {e}+x-x^4\right )^2}-\frac {1}{4 e \left (3+i \sqrt {e}-x+x^4\right )^2}-\frac {1}{2 e \left (-e-\left (3-x+x^4\right )^2\right )}\right ) \, dx\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e}-\frac {(4 (9+e) (i \pi +\log (3))) \int \frac {1}{-e-\left (3-x+x^4\right )^2} \, dx}{e}\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {\sqrt {e}}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{e^2}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {\sqrt {e}}{3 i+\sqrt {e}-i x+i x^4} \, dx}{e^2}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e}\\ &=(6 (i \pi +\log (3))) \int \frac {x^2}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(6 (i \pi +\log (3))) \int \frac {x^5}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx+(24 (i \pi +\log (3))) \int \frac {x^4}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-(42 (i \pi +\log (3))) \int \frac {x}{\left (e+\left (3-x+x^4\right )^2\right )^2} \, dx-\frac {(7 (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{2 \sqrt {e}}-\frac {(7 (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{2 \sqrt {e}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{-3 i+\sqrt {e}+i x-i x^4} \, dx}{e^{3/2}}+\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{3 i+\sqrt {e}-i x+i x^4} \, dx}{e^{3/2}}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (-3+i \sqrt {e}+x-x^4\right )^2} \, dx}{e}-\frac {(2 (9+e) (i \pi +\log (3))) \int \frac {1}{\left (3+i \sqrt {e}-x+x^4\right )^2} \, dx}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 1.00 \begin {gather*} \frac {x (i \pi +\log (3))}{e+\left (3-x+x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((9 + E - x^2 - 18*x^4 + 8*x^5 - 7*x^8)*(I*Pi + Log[3]))/(81 + E^2 - 108*x + 54*x^2 - 12*x^3 + 109*x
^4 - 108*x^5 + 36*x^6 - 4*x^7 + 54*x^8 - 36*x^9 + 6*x^10 + 12*x^12 - 4*x^13 + x^16 + E*(18 - 12*x + 2*x^2 + 12
*x^4 - 4*x^5 + 2*x^8)),x]

[Out]

(x*(I*Pi + Log[3]))/(E + (3 - x + x^4)^2)

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fricas [A]  time = 0.82, size = 35, normalized size = 1.46 \begin {gather*} \frac {i \, \pi x + x \log \relax (3)}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8-4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+
x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm="fr
icas")

[Out]

(I*pi*x + x*log(3))/(x^8 - 2*x^5 + 6*x^4 + x^2 - 6*x + e + 9)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (i \, \pi + \log \relax (3)\right )} {\left (7 \, x^{8} - 8 \, x^{5} + 18 \, x^{4} + x^{2} - e - 9\right )}}{x^{16} - 4 \, x^{13} + 12 \, x^{12} + 6 \, x^{10} - 36 \, x^{9} + 54 \, x^{8} - 4 \, x^{7} + 36 \, x^{6} - 108 \, x^{5} + 109 \, x^{4} - 12 \, x^{3} + 54 \, x^{2} + 2 \, {\left (x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + 9\right )} e - 108 \, x + e^{2} + 81}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8-4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+
x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm="gi
ac")

[Out]

integrate(-(I*pi + log(3))*(7*x^8 - 8*x^5 + 18*x^4 + x^2 - e - 9)/(x^16 - 4*x^13 + 12*x^12 + 6*x^10 - 36*x^9 +
54*x^8 - 4*x^7 + 36*x^6 - 108*x^5 + 109*x^4 - 12*x^3 + 54*x^2 + 2*(x^8 - 2*x^5 + 6*x^4 + x^2 - 6*x + 9)*e - 1
08*x + e^2 + 81), x)

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maple [A]  time = 0.34, size = 35, normalized size = 1.46

method	result	size
gosper	\(\frac {\left (\ln \relax (3)+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\)	\(35\)
norman	\(\frac {\left (\ln \relax (3)+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\)	\(35\)
risch	\(\frac {\left (\ln \relax (3)+i \pi \right ) x}{x^{8}-2 x^{5}+6 x^{4}+x^{2}+{\mathrm e}-6 x +9}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(ln(3)+I*Pi)/(exp(1)^2+(2*x^8-4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+x^16-4*
x^13+12*x^12+6*x^10-36*x^9+54*x^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x,method=_RETURNVERBOSE
)

[Out]

(ln(3)+I*Pi)*x/(x^8-2*x^5+6*x^4+x^2+exp(1)-6*x+9)

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maxima [A]  time = 0.35, size = 33, normalized size = 1.38 \begin {gather*} \frac {{\left (i \, \pi + \log \relax (3)\right )} x}{x^{8} - 2 \, x^{5} + 6 \, x^{4} + x^{2} - 6 \, x + e + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)-7*x^8+8*x^5-18*x^4-x^2+9)*(log(3)+I*pi)/(exp(1)^2+(2*x^8-4*x^5+12*x^4+2*x^2-12*x+18)*exp(1)+
x^16-4*x^13+12*x^12+6*x^10-36*x^9+54*x^8-4*x^7+36*x^6-108*x^5+109*x^4-12*x^3+54*x^2-108*x+81),x, algorithm="ma
xima")

[Out]

(I*pi + log(3))*x/(x^8 - 2*x^5 + 6*x^4 + x^2 - 6*x + e + 9)

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mupad [B]  time = 13.40, size = 34, normalized size = 1.42 \begin {gather*} \frac {x\,\left (\ln \relax (3)+\Pi \,1{}\mathrm {i}\right )}{x^8-2\,x^5+6\,x^4+x^2-6\,x+\mathrm {e}+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((Pi*1i + log(3))*(exp(1) - x^2 - 18*x^4 + 8*x^5 - 7*x^8 + 9))/(exp(2) - 108*x + exp(1)*(2*x^2 - 12*x + 12
*x^4 - 4*x^5 + 2*x^8 + 18) + 54*x^2 - 12*x^3 + 109*x^4 - 108*x^5 + 36*x^6 - 4*x^7 + 54*x^8 - 36*x^9 + 6*x^10 +
12*x^12 - 4*x^13 + x^16 + 81),x)

[Out]

(x*(Pi*1i + log(3)))/(exp(1) - 6*x + x^2 + 6*x^4 - 2*x^5 + x^8 + 9)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)-7*x**8+8*x**5-18*x**4-x**2+9)*(ln(3)+I*pi)/(exp(1)**2+(2*x**8-4*x**5+12*x**4+2*x**2-12*x+18)
*exp(1)+x**16-4*x**13+12*x**12+6*x**10-36*x**9+54*x**8-4*x**7+36*x**6-108*x**5+109*x**4-12*x**3+54*x**2-108*x+
81),x)

[Out]

Timed out

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