3.96.83 \(\int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} (110-30 x+18 e^{28} x^4+e^{14} (-24 x^2-18 x^3)+e^{-\frac {3 x}{-5+3 e^{14} x^2}} (-25+30 e^{14} x^2-9 e^{28} x^4))}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx\)

Optimal. Leaf size=28 \[ -4-x+e^{-\frac {3 x}{5-3 e^{14} x^2}} (-4+2 x) \]

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Rubi [F]  time = 4.93, 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 {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3*x)/(-5 + 3*E^14*x^2))*(110 - 30*x + 18*E^28*x^4 + E^14*(-24*x^2 - 18*x^3) + (-25 + 30*E^14*x^2 - 9*
E^28*x^4)/E^((3*x)/(-5 + 3*E^14*x^2))))/(25 - 30*E^14*x^2 + 9*E^28*x^4),x]

[Out]

-x + 2*E^28*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2)), x] + (7*E^28*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*
x^2))/(Sqrt[5] - Sqrt[3]*E^7*x), x])/(2*Sqrt[5]) - (3*Sqrt[5]*E^28*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2)
)/(Sqrt[5] - Sqrt[3]*E^7*x), x])/2 + (E^21*(Sqrt[15] + 4*E^7)*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2))/(Sq
rt[5] - Sqrt[3]*E^7*x), x])/Sqrt[5] + (7*E^28*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2))/(Sqrt[5] + Sqrt[3]*
E^7*x), x])/(2*Sqrt[5]) - (3*Sqrt[5]*E^28*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2))/(Sqrt[5] + Sqrt[3]*E^7*
x), x])/2 - (E^21*(Sqrt[15] - 4*E^7)*Defer[Int][E^(-28 + (3*x)/(-5 + 3*E^14*x^2))/(Sqrt[5] + Sqrt[3]*E^7*x), x
])/Sqrt[5] + 18*E^28*Defer[Int][E^(-14 + (3*x)/(-5 + 3*E^14*x^2))/(Sqrt[15]*E^7 - 3*E^14*x)^2, x] + 18*E^28*De
fer[Int][E^(-14 + (3*x)/(-5 + 3*E^14*x^2))/(Sqrt[15]*E^7 + 3*E^14*x)^2, x] - 60*E^28*Defer[Int][(E^(-28 + (3*x
)/(-5 + 3*E^14*x^2))*x)/(-5 + 3*E^14*x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (9 e^{28}\right ) \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{\left (-15 e^{14}+9 e^{28} x^2\right )^2} \, dx\\ &=\left (9 e^{28}\right ) \int \left (-\frac {1}{9} \exp \left (-28+\frac {3 x}{5-3 e^{14} x^2}+\frac {3 x}{-5+3 e^{14} x^2}\right )+\frac {110 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{9 \left (-5+3 e^{14} x^2\right )^2}-\frac {10 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{3 \left (-5+3 e^{14} x^2\right )^2}+\frac {2 e^{\frac {3 x}{-5+3 e^{14} x^2}} x^4}{\left (-5+3 e^{14} x^2\right )^2}-\frac {2 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}} x^2 (4+3 x)}{3 \left (-5+3 e^{14} x^2\right )^2}\right ) \, dx\\ &=-\left (e^{28} \int \exp \left (-28+\frac {3 x}{5-3 e^{14} x^2}+\frac {3 x}{-5+3 e^{14} x^2}\right ) \, dx\right )-\left (6 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}} x^2 (4+3 x)}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\left (18 e^{28}\right ) \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} x^4}{\left (-5+3 e^{14} x^2\right )^2} \, dx-\left (30 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\left (110 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (-5+3 e^{14} x^2\right )^2} \, dx\\ &=-\left (e^{28} \int \frac {1}{e^{28}} \, dx\right )-\left (6 e^{28}\right ) \int \left (\frac {5 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} (4+3 x)}{3 \left (-5+3 e^{14} x^2\right )^2}+\frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} (4+3 x)}{3 \left (-5+3 e^{14} x^2\right )}\right ) \, dx+\left (18 e^{28}\right ) \int \left (\frac {1}{9} e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}+\frac {25 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{9 \left (-5+3 e^{14} x^2\right )^2}+\frac {10 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{9 \left (-5+3 e^{14} x^2\right )}\right ) \, dx-\left (30 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\left (110 e^{28}\right ) \int \left (\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{20 \left (\sqrt {15} e^7-3 e^{14} x\right )^2}+\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{20 \left (\sqrt {15} e^7+3 e^{14} x\right )^2}+\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{10 \left (15 e^{14}-9 e^{28} x^2\right )}\right ) \, dx\\ &=-x+\left (2 e^{28}\right ) \int e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} \, dx-\left (2 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} (4+3 x)}{-5+3 e^{14} x^2} \, dx-\left (10 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} (4+3 x)}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7-3 e^{14} x\right )^2} \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7+3 e^{14} x\right )^2} \, dx+\left (20 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{-5+3 e^{14} x^2} \, dx-\left (30 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{15 e^{14}-9 e^{28} x^2} \, dx+\left (50 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (-5+3 e^{14} x^2\right )^2} \, dx\\ &=-x+\left (2 e^{28}\right ) \int e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} \, dx-\left (2 e^{28}\right ) \int \left (-\frac {\left (4 \sqrt {5}+\frac {5 \sqrt {3}}{e^7}\right ) e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{10 \left (\sqrt {5}-\sqrt {3} e^7 x\right )}-\frac {\left (4 \sqrt {5}-\frac {5 \sqrt {3}}{e^7}\right ) e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{10 \left (\sqrt {5}+\sqrt {3} e^7 x\right )}\right ) \, dx-\left (10 e^{28}\right ) \int \left (\frac {4 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (-5+3 e^{14} x^2\right )^2}+\frac {3 e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2}\right ) \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7-3 e^{14} x\right )^2} \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7+3 e^{14} x\right )^2} \, dx+\left (20 e^{28}\right ) \int \left (-\frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{2 \sqrt {5} \left (\sqrt {5}-\sqrt {3} e^7 x\right )}-\frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{2 \sqrt {5} \left (\sqrt {5}+\sqrt {3} e^7 x\right )}\right ) \, dx-\left (30 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\left (33 e^{28}\right ) \int \left (\frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{6 \sqrt {5} \left (\sqrt {5}-\sqrt {3} e^7 x\right )}+\frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{6 \sqrt {5} \left (\sqrt {5}+\sqrt {3} e^7 x\right )}\right ) \, dx+\left (50 e^{28}\right ) \int \left (\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{20 \left (\sqrt {15} e^7-3 e^{14} x\right )^2}+\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{20 \left (\sqrt {15} e^7+3 e^{14} x\right )^2}+\frac {3 e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{10 \left (15 e^{14}-9 e^{28} x^2\right )}\right ) \, dx\\ &=-x+\left (2 e^{28}\right ) \int e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} \, dx+\frac {1}{2} \left (15 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7-3 e^{14} x\right )^2} \, dx+\frac {1}{2} \left (15 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7+3 e^{14} x\right )^2} \, dx+\left (15 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{15 e^{14}-9 e^{28} x^2} \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7-3 e^{14} x\right )^2} \, dx+\frac {1}{2} \left (33 e^{28}\right ) \int \frac {e^{-14+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (\sqrt {15} e^7+3 e^{14} x\right )^2} \, dx-2 \left (\left (30 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}} x}{\left (-5+3 e^{14} x^2\right )^2} \, dx\right )-\left (40 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\left (-5+3 e^{14} x^2\right )^2} \, dx+\frac {\left (11 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}-\sqrt {3} e^7 x} \, dx}{2 \sqrt {5}}+\frac {\left (11 e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}+\sqrt {3} e^7 x} \, dx}{2 \sqrt {5}}-\left (2 \sqrt {5} e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}-\sqrt {3} e^7 x} \, dx-\left (2 \sqrt {5} e^{28}\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}+\sqrt {3} e^7 x} \, dx-\frac {\left (e^{21} \left (\sqrt {15}-4 e^7\right )\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}+\sqrt {3} e^7 x} \, dx}{\sqrt {5}}+\frac {\left (e^{21} \left (\sqrt {15}+4 e^7\right )\right ) \int \frac {e^{-28+\frac {3 x}{-5+3 e^{14} x^2}}}{\sqrt {5}-\sqrt {3} e^7 x} \, dx}{\sqrt {5}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 27, normalized size = 0.96 \begin {gather*} -x+e^{\frac {3 x}{-5+3 e^{14} x^2}} (-4+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*x)/(-5 + 3*E^14*x^2))*(110 - 30*x + 18*E^28*x^4 + E^14*(-24*x^2 - 18*x^3) + (-25 + 30*E^14*x^
2 - 9*E^28*x^4)/E^((3*x)/(-5 + 3*E^14*x^2))))/(25 - 30*E^14*x^2 + 9*E^28*x^4),x]

[Out]

-x + E^((3*x)/(-5 + 3*E^14*x^2))*(-4 + 2*x)

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fricas [A]  time = 0.64, size = 24, normalized size = 0.86 \begin {gather*} 2 \, {\left (x - 2\right )} e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*
exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="fricas")

[Out]

2*(x - 2)*e^(3*x/(3*x^2*e^14 - 5)) - x

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giac [A]  time = 0.25, size = 39, normalized size = 1.39 \begin {gather*} 2 \, x e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} - x - 4 \, e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*
exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="giac")

[Out]

2*x*e^(3*x/(3*x^2*e^14 - 5)) - x - 4*e^(3*x/(3*x^2*e^14 - 5))

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maple [A]  time = 0.56, size = 26, normalized size = 0.93

method	result	size
risch	\(-x +\left (2 x -4\right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}\)	\(26\)
norman	\(\frac {\left (20-10 x +5 x \,{\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}-12 x^{2} {\mathrm e}^{14}+6 \,{\mathrm e}^{14} x^{3}-3 \,{\mathrm e}^{14} x^{3} {\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}\right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}}{3 x^{2} {\mathrm e}^{14}-5}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)
^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x,method=_RETURNVERBOSE)

[Out]

-x+(2*x-4)*exp(3*x/(3*x^2*exp(14)-5))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{12} \, \sqrt {15} e^{\left (-7\right )} \log \left (\frac {3 \, x e^{14} - \sqrt {15} e^{7}}{3 \, x e^{14} + \sqrt {15} e^{7}}\right ) - \frac {1}{4} \, {\left (\sqrt {15} e^{\left (-35\right )} \log \left (\frac {3 \, x e^{14} - \sqrt {15} e^{7}}{3 \, x e^{14} + \sqrt {15} e^{7}}\right ) + 4 \, x e^{\left (-28\right )} - \frac {10 \, x}{3 \, x^{2} e^{42} - 5 \, e^{28}}\right )} e^{28} + \frac {1}{6} \, {\left (\sqrt {15} e^{\left (-21\right )} \log \left (\frac {3 \, x e^{14} - \sqrt {15} e^{7}}{3 \, x e^{14} + \sqrt {15} e^{7}}\right ) - \frac {30 \, x}{3 \, x^{2} e^{28} - 5 \, e^{14}}\right )} e^{14} + 2 \, x e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} + \frac {5 \, x}{2 \, {\left (3 \, x^{2} e^{14} - 5\right )}} + \int \frac {12 \, {\left (3 \, x^{2} e^{14} + 5\right )} e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )}}{9 \, x^{4} e^{28} - 30 \, x^{2} e^{14} + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*
exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="maxima")

[Out]

1/12*sqrt(15)*e^(-7)*log((3*x*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt(15)*e^7)) - 1/4*(sqrt(15)*e^(-35)*log((3*x
*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt(15)*e^7)) + 4*x*e^(-28) - 10*x/(3*x^2*e^42 - 5*e^28))*e^28 + 1/6*(sqrt(
15)*e^(-21)*log((3*x*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt(15)*e^7)) - 30*x/(3*x^2*e^28 - 5*e^14))*e^14 + 2*x*
e^(3*x/(3*x^2*e^14 - 5)) + 5/2*x/(3*x^2*e^14 - 5) + integrate(12*(3*x^2*e^14 + 5)*e^(3*x/(3*x^2*e^14 - 5))/(9*
x^4*e^28 - 30*x^2*e^14 + 25), x)

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mupad [B]  time = 9.32, size = 25, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{\frac {3\,x}{3\,x^2\,{\mathrm {e}}^{14}-5}}\,\left (2\,x-4\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((3*x)/(3*x^2*exp(14) - 5))*(30*x + exp(-(3*x)/(3*x^2*exp(14) - 5))*(9*x^4*exp(28) - 30*x^2*exp(14) +
25) + exp(14)*(24*x^2 + 18*x^3) - 18*x^4*exp(28) - 110))/(9*x^4*exp(28) - 30*x^2*exp(14) + 25),x)

[Out]

exp((3*x)/(3*x^2*exp(14) - 5))*(2*x - 4) - x

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sympy [A]  time = 0.54, size = 20, normalized size = 0.71 \begin {gather*} - x + \left (2 x - 4\right ) e^{\frac {3 x}{3 x^{2} e^{14} - 5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**4*exp(7)**4+30*x**2*exp(7)**2-25)*exp(-3*x/(3*x**2*exp(7)**2-5))+18*x**4*exp(7)**4+(-18*x**3
-24*x**2)*exp(7)**2-30*x+110)/(9*x**4*exp(7)**4-30*x**2*exp(7)**2+25)/exp(-3*x/(3*x**2*exp(7)**2-5)),x)

[Out]

-x + (2*x - 4)*exp(3*x/(3*x**2*exp(14) - 5))

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