3.100.29 \(\int \frac {-290 x+e^{x^3} (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5)+e^{2 x^3} (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6)}{4+4 x^2+x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac {5 \left (4+\left (5-e^{x^3} (-6+x)\right )^2\right )}{2+x^2} \]

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Rubi [F]  time = 1.66, 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 {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-290*x + E^x^3*(-100 - 600*x + 1850*x^2 - 300*x^3 + 900*x^4 - 150*x^5) + E^(2*x^3)*(-120 - 340*x + 2220*x
^2 - 720*x^3 + 1140*x^4 - 360*x^5 + 30*x^6))/(4 + 4*x^2 + x^4),x]

[Out]

(25*E^x^3)/(I*Sqrt[2] - x) - (25*E^x^3)/(I*Sqrt[2] + x) + 145/(2 + x^2) + (5*E^(2*x^3)*(6 - x)*(12*x^2 - 2*x^3
+ 6*x^4 - x^5))/(x^2*(2 + x^2)^2) - (300*x*Gamma[1/3, -x^3])/(-x^3)^(1/3) + 150*Defer[Int][E^x^3/(I*Sqrt[2] -
x), x] - ((25*I)*Defer[Int][E^x^3/(I*Sqrt[2] - x), x])/Sqrt[2] - (25*(12 + (35*I)*Sqrt[2])*Defer[Int][E^x^3/(
I*Sqrt[2] - x), x])/2 - 150*Defer[Int][E^x^3/(I*Sqrt[2] + x), x] - ((25*I)*Defer[Int][E^x^3/(I*Sqrt[2] + x), x
])/Sqrt[2] + (25*(12 - (35*I)*Sqrt[2])*Defer[Int][E^x^3/(I*Sqrt[2] + x), x])/2 - 600*Defer[Int][(E^x^3*x)/(2 +
x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{\left (2+x^2\right )^2} \, dx\\ &=\int \left (-\frac {290 x}{\left (2+x^2\right )^2}-\frac {50 e^{x^3} \left (2+12 x-37 x^2+6 x^3-18 x^4+3 x^5\right )}{\left (2+x^2\right )^2}+\frac {10 e^{2 x^3} (-6+x) \left (2+6 x-36 x^2+6 x^3-18 x^4+3 x^5\right )}{\left (2+x^2\right )^2}\right ) \, dx\\ &=10 \int \frac {e^{2 x^3} (-6+x) \left (2+6 x-36 x^2+6 x^3-18 x^4+3 x^5\right )}{\left (2+x^2\right )^2} \, dx-50 \int \frac {e^{x^3} \left (2+12 x-37 x^2+6 x^3-18 x^4+3 x^5\right )}{\left (2+x^2\right )^2} \, dx-290 \int \frac {x}{\left (2+x^2\right )^2} \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-50 \int \left (-18 e^{x^3}+3 e^{x^3} x+\frac {4 e^{x^3} (1+3 x)}{\left (2+x^2\right )^2}+\frac {e^{x^3} (35-6 x)}{2+x^2}\right ) \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-50 \int \frac {e^{x^3} (35-6 x)}{2+x^2} \, dx-150 \int e^{x^3} x \, dx-200 \int \frac {e^{x^3} (1+3 x)}{\left (2+x^2\right )^2} \, dx+900 \int e^{x^3} \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}-50 \int \left (\frac {\left (12+35 i \sqrt {2}\right ) e^{x^3}}{4 \left (i \sqrt {2}-x\right )}+\frac {\left (-12+35 i \sqrt {2}\right ) e^{x^3}}{4 \left (i \sqrt {2}+x\right )}\right ) \, dx-200 \int \left (\frac {e^{x^3}}{\left (2+x^2\right )^2}+\frac {3 e^{x^3} x}{\left (2+x^2\right )^2}\right ) \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}-200 \int \frac {e^{x^3}}{\left (2+x^2\right )^2} \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}-200 \int \left (-\frac {e^{x^3}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{x^3}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{x^3}}{4 \left (-2-x^2\right )}\right ) \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}+25 \int \frac {e^{x^3}}{\left (i \sqrt {2}-x\right )^2} \, dx+25 \int \frac {e^{x^3}}{\left (i \sqrt {2}+x\right )^2} \, dx+50 \int \frac {e^{x^3}}{-2-x^2} \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {25 e^{x^3}}{i \sqrt {2}-x}-\frac {25 e^{x^3}}{i \sqrt {2}+x}+\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}+50 \int \left (-\frac {i e^{x^3}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{x^3}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx-75 \int \frac {e^{x^3} x^2}{i \sqrt {2}-x} \, dx+75 \int \frac {e^{x^3} x^2}{i \sqrt {2}+x} \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {25 e^{x^3}}{i \sqrt {2}-x}-\frac {25 e^{x^3}}{i \sqrt {2}+x}+\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}-75 \int \left (-i \sqrt {2} e^{x^3}-\frac {2 e^{x^3}}{i \sqrt {2}-x}-e^{x^3} x\right ) \, dx+75 \int \left (-i \sqrt {2} e^{x^3}+e^{x^3} x-\frac {2 e^{x^3}}{i \sqrt {2}+x}\right ) \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx}{\sqrt {2}}-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx}{\sqrt {2}}+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {25 e^{x^3}}{i \sqrt {2}-x}-\frac {25 e^{x^3}}{i \sqrt {2}+x}+\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+\frac {50 x^2 \Gamma \left (\frac {2}{3},-x^3\right )}{\left (-x^3\right )^{2/3}}+2 \left (75 \int e^{x^3} x \, dx\right )+150 \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx-150 \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx}{\sqrt {2}}-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx}{\sqrt {2}}+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ &=\frac {25 e^{x^3}}{i \sqrt {2}-x}-\frac {25 e^{x^3}}{i \sqrt {2}+x}+\frac {145}{2+x^2}+\frac {5 e^{2 x^3} (6-x) \left (12 x^2-2 x^3+6 x^4-x^5\right )}{x^2 \left (2+x^2\right )^2}-\frac {300 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}+150 \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx-150 \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-600 \int \frac {e^{x^3} x}{\left (2+x^2\right )^2} \, dx-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx}{\sqrt {2}}-\frac {(25 i) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx}{\sqrt {2}}+\frac {1}{2} \left (25 \left (12-35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}+x} \, dx-\frac {1}{2} \left (25 \left (12+35 i \sqrt {2}\right )\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 34, normalized size = 1.36 \begin {gather*} \frac {5 \left (29-10 e^{x^3} (-6+x)+e^{2 x^3} (-6+x)^2\right )}{2+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-290*x + E^x^3*(-100 - 600*x + 1850*x^2 - 300*x^3 + 900*x^4 - 150*x^5) + E^(2*x^3)*(-120 - 340*x +
2220*x^2 - 720*x^3 + 1140*x^4 - 360*x^5 + 30*x^6))/(4 + 4*x^2 + x^4),x]

[Out]

(5*(29 - 10*E^x^3*(-6 + x) + E^(2*x^3)*(-6 + x)^2))/(2 + x^2)

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fricas [A]  time = 0.56, size = 35, normalized size = 1.40 \begin {gather*} \frac {5 \, {\left ({\left (x^{2} - 12 \, x + 36\right )} e^{\left (2 \, x^{3}\right )} - 10 \, {\left (x - 6\right )} e^{\left (x^{3}\right )} + 29\right )}}{x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150*x^5+900*x^4-300*x^3+1850*x^2-
600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x, algorithm="fricas")

[Out]

5*((x^2 - 12*x + 36)*e^(2*x^3) - 10*(x - 6)*e^(x^3) + 29)/(x^2 + 2)

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giac [B]  time = 1.23, size = 51, normalized size = 2.04 \begin {gather*} \frac {5 \, {\left (x^{2} e^{\left (2 \, x^{3}\right )} - 12 \, x e^{\left (2 \, x^{3}\right )} - 10 \, x e^{\left (x^{3}\right )} + 36 \, e^{\left (2 \, x^{3}\right )} + 60 \, e^{\left (x^{3}\right )} + 29\right )}}{x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150*x^5+900*x^4-300*x^3+1850*x^2-
600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x, algorithm="giac")

[Out]

5*(x^2*e^(2*x^3) - 12*x*e^(2*x^3) - 10*x*e^(x^3) + 36*e^(2*x^3) + 60*e^(x^3) + 29)/(x^2 + 2)

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maple [B]  time = 0.15, size = 50, normalized size = 2.00

method	result	size
risch	\(\frac {145}{x^{2}+2}+\frac {5 \left (x^{2}-12 x +36\right ) {\mathrm e}^{2 x^{3}}}{x^{2}+2}-\frac {50 \left (x -6\right ) {\mathrm e}^{x^{3}}}{x^{2}+2}\)	\(50\)
norman	\(\frac {180 \,{\mathrm e}^{2 x^{3}}+5 \,{\mathrm e}^{2 x^{3}} x^{2}-50 \,{\mathrm e}^{x^{3}} x -60 \,{\mathrm e}^{2 x^{3}} x +300 \,{\mathrm e}^{x^{3}}+145}{x^{2}+2}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150*x^5+900*x^4-300*x^3+1850*x^2-600*x-
100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x,method=_RETURNVERBOSE)

[Out]

145/(x^2+2)+5*(x^2-12*x+36)/(x^2+2)*exp(2*x^3)-50*(x-6)/(x^2+2)*exp(x^3)

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maxima [A]  time = 0.39, size = 44, normalized size = 1.76 \begin {gather*} \frac {5 \, {\left ({\left (x^{2} - 12 \, x + 36\right )} e^{\left (2 \, x^{3}\right )} - 10 \, {\left (x - 6\right )} e^{\left (x^{3}\right )}\right )}}{x^{2} + 2} + \frac {145}{x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150*x^5+900*x^4-300*x^3+1850*x^2-
600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x, algorithm="maxima")

[Out]

5*((x^2 - 12*x + 36)*e^(2*x^3) - 10*(x - 6)*e^(x^3))/(x^2 + 2) + 145/(x^2 + 2)

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mupad [B]  time = 6.82, size = 51, normalized size = 2.04 \begin {gather*} 5\,{\mathrm {e}}^{2\,x^3}+\frac {300\,{\mathrm {e}}^{x^3}+170\,{\mathrm {e}}^{2\,x^3}-x\,\left (50\,{\mathrm {e}}^{x^3}+60\,{\mathrm {e}}^{2\,x^3}\right )+145}{x^2+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(290*x + exp(2*x^3)*(340*x - 2220*x^2 + 720*x^3 - 1140*x^4 + 360*x^5 - 30*x^6 + 120) + exp(x^3)*(600*x -
1850*x^2 + 300*x^3 - 900*x^4 + 150*x^5 + 100))/(4*x^2 + x^4 + 4),x)

[Out]

5*exp(2*x^3) + (300*exp(x^3) + 170*exp(2*x^3) - x*(50*exp(x^3) + 60*exp(2*x^3)) + 145)/(x^2 + 2)

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sympy [B]  time = 0.42, size = 65, normalized size = 2.60 \begin {gather*} \frac {\left (- 50 x^{3} + 300 x^{2} - 100 x + 600\right ) e^{x^{3}} + \left (5 x^{4} - 60 x^{3} + 190 x^{2} - 120 x + 360\right ) e^{2 x^{3}}}{x^{4} + 4 x^{2} + 4} + \frac {290}{2 x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x**6-360*x**5+1140*x**4-720*x**3+2220*x**2-340*x-120)*exp(x**3)**2+(-150*x**5+900*x**4-300*x**3
+1850*x**2-600*x-100)*exp(x**3)-290*x)/(x**4+4*x**2+4),x)

[Out]

((-50*x**3 + 300*x**2 - 100*x + 600)*exp(x**3) + (5*x**4 - 60*x**3 + 190*x**2 - 120*x + 360)*exp(2*x**3))/(x**
4 + 4*x**2 + 4) + 290/(2*x**2 + 4)

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