3.59.19 \(\int \frac {-5 x^2+10 x^4+e^{x^2} (75-24 x^2)+(5 x^2+e^{x^2} (75+24 x^2)) \log (\frac {e^{-x^2} (-5 x^2+e^{x^2} (-75-24 x^2))}{5 x})}{(5 x^2+e^{x^2} (75+24 x^2)) \log ^2(\frac {e^{-x^2} (-5 x^2+e^{x^2} (-75-24 x^2))}{5 x})} \, dx\)

Optimal. Leaf size=35 \[ \frac {x}{\log \left (x \left (\frac {1}{5}-\frac {e^{-x^2} x+5 \left (\frac {3}{x}+x\right )}{x}\right )\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 5.31, 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 {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x^2 + 10*x^4 + E^x^2*(75 - 24*x^2) + (5*x^2 + E^x^2*(75 + 24*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))
/(5*E^x^2*x)])/((5*x^2 + E^x^2*(75 + 24*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)]^2),x]

[Out]

-Defer[Int][Log[-15/x + (-24/5 - E^(-x^2))*x]^(-2), x] + (5*I)*Defer[Int][1/((5*I - 2*Sqrt[2]*x)*Log[-15/x + (
-24/5 - E^(-x^2))*x]^2), x] + (5*I)*Defer[Int][1/((5*I + 2*Sqrt[2]*x)*Log[-15/x + (-24/5 - E^(-x^2))*x]^2), x]
- (125*Defer[Int][1/((75*E^x^2 + 5*x^2 + 24*E^x^2*x^2)*Log[-15/x + (-24/5 - E^(-x^2))*x]^2), x])/4 + 10*Defer
[Int][x^4/((75*E^x^2 + 5*x^2 + 24*E^x^2*x^2)*Log[-15/x + (-24/5 - E^(-x^2))*x]^2), x] + ((625*I)/8)*Defer[Int]
[1/((5*I - 2*Sqrt[2]*x)*(75*E^x^2 + 5*x^2 + 24*E^x^2*x^2)*Log[-15/x + (-24/5 - E^(-x^2))*x]^2), x] + ((625*I)/
8)*Defer[Int][1/((5*I + 2*Sqrt[2]*x)*(75*E^x^2 + 5*x^2 + 24*E^x^2*x^2)*Log[-15/x + (-24/5 - E^(-x^2))*x]^2), x
] + Defer[Int][Log[-15/x + (-24/5 - E^(-x^2))*x]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {10 x^2 \left (-25+25 x^2+8 x^4\right )}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {25-8 x^2+25 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )+8 x^2 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx\\ &=10 \int \frac {x^2 \left (-25+25 x^2+8 x^4\right )}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {25-8 x^2+25 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )+8 x^2 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=10 \int \left (-\frac {25}{8 \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {625}{8 \left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {25-8 x^2+\left (25+8 x^2\right ) \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {3125}{4} \int \frac {1}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \left (\frac {25-8 x^2}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx\\ &=10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {3125}{4} \int \left (\frac {i}{10 \left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {i}{10 \left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {25-8 x^2}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \left (-\frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {50}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+50 \int \frac {1}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+50 \int \left (\frac {i}{10 \left (5 i-2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {i}{10 \left (5 i+2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=5 i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+5 i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 26, normalized size = 0.74 \begin {gather*} \frac {x}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x^2 + 10*x^4 + E^x^2*(75 - 24*x^2) + (5*x^2 + E^x^2*(75 + 24*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24
*x^2))/(5*E^x^2*x)])/((5*x^2 + E^x^2*(75 + 24*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)]^2),x]

[Out]

x/Log[-15/x + (-24/5 - E^(-x^2))*x]

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 35, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (-\frac {{\left (5 \, x^{2} + 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2
)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm=
"fricas")

[Out]

x/log(-1/5*(5*x^2 + 3*(8*x^2 + 25)*e^(x^2))*e^(-x^2)/x)

________________________________________________________________________________________

giac [A]  time = 0.62, size = 37, normalized size = 1.06 \begin {gather*} \frac {x}{\log \left (-\frac {{\left (24 \, x^{2} e^{\left (x^{2}\right )} + 5 \, x^{2} + 75 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2
)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm=
"giac")

[Out]

x/log(-1/5*(24*x^2*e^(x^2) + 5*x^2 + 75*e^(x^2))*e^(-x^2)/x)

________________________________________________________________________________________

maple [C]  time = 0.24, size = 486, normalized size = 13.89

method	result	size
risch	\(\frac {2 i x}{2 \pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )-\pi \,\mathrm {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-\pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 \pi +2 i \ln \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )-2 i \ln \relax (x )-2 i \ln \relax (5)-2 i \ln \left ({\mathrm e}^{x^{2}}\right )+2 i \ln \relax (3)+6 i \ln \relax (2)}\)	\(486\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((24*x^2+75)*exp(x^2)+5*x^2)*ln(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^
4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2)/ln(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x,method=_RETURNVERBO
SE)

[Out]

2*I*x/(2*Pi*csgn(I/x*((exp(x^2)+5/24)*x^2+25/8*exp(x^2))*exp(-x^2))^2+Pi*csgn(I*((exp(x^2)+5/24)*x^2+25/8*exp(
x^2)))*csgn(I*exp(-x^2))*csgn(I*exp(-x^2)*((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))-Pi*csgn(I*((exp(x^2)+5/24)*x^2+
25/8*exp(x^2)))*csgn(I*exp(-x^2)*((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))^2-Pi*csgn(I*exp(-x^2))*csgn(I*exp(-x^2)*
((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))^2+Pi*csgn(I*exp(-x^2)*((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))^3-Pi*csgn(I*ex
p(-x^2)*((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))*csgn(I/x*((exp(x^2)+5/24)*x^2+25/8*exp(x^2))*exp(-x^2))^2+Pi*csgn
(I*exp(-x^2)*((exp(x^2)+5/24)*x^2+25/8*exp(x^2)))*csgn(I/x*((exp(x^2)+5/24)*x^2+25/8*exp(x^2))*exp(-x^2))*csgn
(I/x)-Pi*csgn(I/x*((exp(x^2)+5/24)*x^2+25/8*exp(x^2))*exp(-x^2))^3-Pi*csgn(I/x*((exp(x^2)+5/24)*x^2+25/8*exp(x
^2))*exp(-x^2))^2*csgn(I/x)-2*Pi+2*I*ln((exp(x^2)+5/24)*x^2+25/8*exp(x^2))-2*I*ln(x)-2*I*ln(5)-2*I*ln(exp(x^2)
)+2*I*ln(3)+6*I*ln(2))

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 35, normalized size = 1.00 \begin {gather*} -\frac {x}{x^{2} + \log \relax (5) - \log \left (-5 \, x^{2} - 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2
)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm=
"maxima")

[Out]

-x/(x^2 + log(5) - log(-5*x^2 - 3*(8*x^2 + 25)*e^(x^2)) + log(x))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2-75\right )-\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )+5\,x^2-10\,x^4}{{\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )}^2\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(24*x^2 - 75) - log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)*(exp(x^2)*(24*x^2 + 75)
+ 5*x^2) + 5*x^2 - 10*x^4)/(log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)^2*(exp(x^2)*(24*x^2 + 75) +
5*x^2)),x)

[Out]

-int((exp(x^2)*(24*x^2 - 75) - log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)*(exp(x^2)*(24*x^2 + 75)
+ 5*x^2) + 5*x^2 - 10*x^4)/(log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)^2*(exp(x^2)*(24*x^2 + 75) +
5*x^2)), x)

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 27, normalized size = 0.77 \begin {gather*} \frac {x}{\log {\left (\frac {\left (- x^{2} + \frac {\left (- 24 x^{2} - 75\right ) e^{x^{2}}}{5}\right ) e^{- x^{2}}}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((24*x**2+75)*exp(x**2)+5*x**2)*ln(1/5*((-24*x**2-75)*exp(x**2)-5*x**2)/exp(x**2)/x)+(-24*x**2+75)*
exp(x**2)+10*x**4-5*x**2)/((24*x**2+75)*exp(x**2)+5*x**2)/ln(1/5*((-24*x**2-75)*exp(x**2)-5*x**2)/exp(x**2)/x)
**2,x)

[Out]

x/log((-x**2 + (-24*x**2 - 75)*exp(x**2)/5)*exp(-x**2)/x)

________________________________________________________________________________________