Integrand size = 11, antiderivative size = 29 \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\frac {1}{3} a^3 \Gamma \left (-\frac {5}{2},a x\right )-\frac {\Gamma \left (\frac {1}{2},a x\right )}{3 x^3} \] Output:
1/3*a^3*(-8/15*Pi^(1/2)*erfc((a*x)^(1/2))+8/15/(a*x)^(1/2)*exp(-a*x)-4/15/ (a*x)^(3/2)*exp(-a*x)+2/5/(a*x)^(5/2)*exp(-a*x))-1/3*Pi^(1/2)*erfc((a*x)^( 1/2))/x^3
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\frac {1}{3} a^3 \Gamma \left (-\frac {5}{2},a x\right )-\frac {\Gamma \left (\frac {1}{2},a x\right )}{3 x^3} \] Input:
Integrate[Gamma[1/2, a*x]/x^4,x]
Output:
(a^3*Gamma[-5/2, a*x])/3 - Gamma[1/2, a*x]/(3*x^3)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7116}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 7116 |
\(\displaystyle \frac {1}{3} a^3 \Gamma \left (-\frac {5}{2},a x\right )-\frac {\Gamma \left (\frac {1}{2},a x\right )}{3 x^3}\) |
Input:
Int[Gamma[1/2, a*x]/x^4,x]
Output:
(a^3*Gamma[-5/2, a*x])/3 - Gamma[1/2, a*x]/(3*x^3)
Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b *(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48
method | result | size |
parts | \(-\frac {\sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right )}{3 x^{3}}-\frac {2 a^{3} \left (-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}\right )}{3}\) | \(72\) |
derivativedivides | \(2 \sqrt {\pi }\, a^{3} \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{6 x^{3} a^{3}}-\frac {-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}}{3 \sqrt {\pi }}\right )\) | \(80\) |
default | \(2 \sqrt {\pi }\, a^{3} \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{6 x^{3} a^{3}}-\frac {-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}}{3 \sqrt {\pi }}\right )\) | \(80\) |
Input:
int(Pi^(1/2)*erfc((x*a)^(1/2))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*Pi^(1/2)*erfc((x*a)^(1/2))/x^3-2/3*a^3*(-1/5/exp(x*a)/(x*a)^(5/2)+2/1 5/exp(x*a)/(x*a)^(3/2)-4/15/exp(x*a)/(x*a)^(1/2)-4/15*Pi^(1/2)*erf((x*a)^( 1/2)))
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\frac {\sqrt {\pi } {\left (8 \, a^{3} x^{3} + 15\right )} \operatorname {erf}\left (\sqrt {a x}\right ) + 2 \, {\left (4 \, a^{2} x^{2} - 2 \, a x + 3\right )} \sqrt {a x} e^{\left (-a x\right )} - 15 \, \sqrt {\pi }}{45 \, x^{3}} \] Input:
integrate(pi^(1/2)*erfc((a*x)^(1/2))/x^4,x, algorithm="fricas")
Output:
1/45*(sqrt(pi)*(8*a^3*x^3 + 15)*erf(sqrt(a*x)) + 2*(4*a^2*x^2 - 2*a*x + 3) *sqrt(a*x)*e^(-a*x) - 15*sqrt(pi))/x^3
Time = 0.69 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69 \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\sqrt {\pi } \left (- \frac {8 a^{3} \operatorname {erfc}{\left (\sqrt {a x} \right )}}{45} + \frac {8 a^{2} \sqrt {a x} e^{- a x}}{45 \sqrt {\pi } x} - \frac {4 a \sqrt {a x} e^{- a x}}{45 \sqrt {\pi } x^{2}} + \frac {2 \sqrt {a x} e^{- a x}}{15 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfc}{\left (\sqrt {a x} \right )}}{3 x^{3}}\right ) \] Input:
integrate(pi**(1/2)*erfc((a*x)**(1/2))/x**4,x)
Output:
sqrt(pi)*(-8*a**3*erfc(sqrt(a*x))/45 + 8*a**2*sqrt(a*x)*exp(-a*x)/(45*sqrt (pi)*x) - 4*a*sqrt(a*x)*exp(-a*x)/(45*sqrt(pi)*x**2) + 2*sqrt(a*x)*exp(-a* x)/(15*sqrt(pi)*x**3) - erfc(sqrt(a*x))/(3*x**3))
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\frac {1}{3} \, \sqrt {\pi } a^{3} {\left (\frac {\Gamma \left (-\frac {5}{2}, a x\right )}{\sqrt {\pi }} - \frac {\operatorname {erfc}\left (\sqrt {a x}\right )}{a^{3} x^{3}}\right )} \] Input:
integrate(pi^(1/2)*erfc((a*x)^(1/2))/x^4,x, algorithm="maxima")
Output:
1/3*sqrt(pi)*a^3*(gamma(-5/2, a*x)/sqrt(pi) - erfc(sqrt(a*x))/(a^3*x^3))
\[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right )}{x^{4}} \,d x } \] Input:
integrate(pi^(1/2)*erfc((a*x)^(1/2))/x^4,x, algorithm="giac")
Output:
integrate(sqrt(pi)*erfc(sqrt(a*x))/x^4, x)
Timed out. \[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\int \frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{x^4} \,d x \] Input:
int((Pi^(1/2)*erfc((a*x)^(1/2)))/x^4,x)
Output:
int((Pi^(1/2)*erfc((a*x)^(1/2)))/x^4, x)
\[ \int \frac {\Gamma \left (\frac {1}{2},a x\right )}{x^4} \, dx=\frac {15 \sqrt {x}\, \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )-4 \sqrt {x}\, e^{a x} \sqrt {a}\, \left (\int \frac {\sqrt {x}}{e^{a x} x^{2}}d x \right ) a^{2} x^{3}-4 \sqrt {a}\, a \,x^{2}+6 \sqrt {a}\, x -15 \sqrt {x}\, \sqrt {\pi }\, e^{a x}}{45 \sqrt {x}\, e^{a x} x^{3}} \] Input:
int(Pi^(1/2)*erfc((a*x)^(1/2))/x^4,x)
Output:
(15*sqrt(x)*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a)) - 4*sqrt(x)*e**(a*x)*sq rt(a)*int(sqrt(x)/(e**(a*x)*x**2),x)*a**2*x**3 - 4*sqrt(a)*a*x**2 + 6*sqrt (a)*x - 15*sqrt(x)*sqrt(pi)*e**(a*x))/(45*sqrt(x)*e**(a*x)*x**3)