Integrand size = 8, antiderivative size = 45 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=b x \, _3F_3(1,1,1;2,2,2;b x)+\gamma \log (x)+(\operatorname {ExpIntegralE}(1,-b x)+\operatorname {ExpIntegralEi}(b x)) \log (x)+\frac {1}{2} \log ^2(-b x) \] Output:
b*x*hypergeom([1, 1, 1],[2, 2, 2],b*x)+gamma*ln(x)+(Ei(1,-b*x)+Ei(b*x))*ln (x)+1/2*ln(-b*x)^2
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=b x \, _3F_3(1,1,1;2,2,2;b x)+\frac {1}{2} \log (x) (2 \gamma +2 \operatorname {ExpIntegralEi}(b x)+2 \Gamma (0,-b x)-\log (x)+2 \log (-b x)) \] Input:
Integrate[ExpIntegralEi[b*x]/x,x]
Output:
b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + (Log[x]*(2*EulerGamma + 2*ExpIntegralEi[b*x] + 2*Gamma[0, -(b*x)] - Log[x] + 2*Log[-(b*x)]))/2
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {7037, 7029}
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.
\(\displaystyle \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx\) |
\(\Big \downarrow \) 7037 |
\(\displaystyle \log (x) (\operatorname {ExpIntegralE}(1,-b x)+\operatorname {ExpIntegralEi}(b x))-\int \frac {\operatorname {ExpIntegralE}(1,-b x)}{x}dx\) |
\(\Big \downarrow \) 7029 |
\(\displaystyle b x \, _3F_3(1,1,1;2,2,2;b x)+\log (x) (\operatorname {ExpIntegralE}(1,-b x)+\operatorname {ExpIntegralEi}(b x))+\frac {1}{2} \log ^2(-b x)+\gamma \log (x)\) |
Input:
Int[ExpIntegralEi[b*x]/x,x]
Output:
b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + EulerGamma*Log[x] + (Ex pIntegralE[1, -(b*x)] + ExpIntegralEi[b*x])*Log[x] + Log[-(b*x)]^2/2
Int[ExpIntegralE[1, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[b*x*HypergeometricP FQ[{1, 1, 1}, {2, 2, 2}, (-b)*x], x] + (-Simp[EulerGamma*Log[x], x] - Simp[ (1/2)*Log[b*x]^2, x]) /; FreeQ[b, x]
Int[ExpIntegralEi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x]*(ExpIntegralEi [b*x] + ExpIntegralE[1, (-b)*x]), x] - Int[ExpIntegralE[1, (-b)*x]/x, x] /; FreeQ[b, x]
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29
method | result | size |
meijerg | \(\ln \left (x \right ) \ln \left (-b \right )+\frac {\gamma ^{2}}{2}+\ln \left (-b \right ) \gamma +\gamma \ln \left (x \right )+\frac {\pi ^{2}}{12}+\frac {\ln \left (x \right )^{2}}{2}+\frac {\ln \left (-b \right )^{2}}{2}+b x \operatorname {hypergeom}\left (\left [1, 1, 1\right ], \left [2, 2, 2\right ], b x \right )\) | \(58\) |
Input:
int(Ei(b*x)/x,x,method=_RETURNVERBOSE)
Output:
ln(x)*ln(-b)+1/2*gamma^2+ln(-b)*gamma+gamma*ln(x)+1/12*Pi^2+1/2*ln(x)^2+1/ 2*ln(-b)^2+b*x*hypergeom([1,1,1],[2,2,2],b*x)
\[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\int { \frac {{\rm Ei}\left (b x\right )}{x} \,d x } \] Input:
integrate(Ei(b*x)/x,x, algorithm="fricas")
Output:
integral(Ei(b*x)/x, x)
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.00 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\begin {cases} b x {{}_{3}F_{3}\left (\begin {matrix} 1, 1, 1 \\ 2, 2, 2 \end {matrix}\middle | {b x} \right )} + i \pi \log {\left (\frac {1}{x} \right )} - i \pi \log {\left (x \right )} + \frac {\log {\left (b x \right )}^{2}}{2} + \gamma \log {\left (b x \right )} + i \pi \log {\left (b x \right )} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\b x {{}_{3}F_{3}\left (\begin {matrix} 1, 1, 1 \\ 2, 2, 2 \end {matrix}\middle | {b x} \right )} - i \pi \log {\left (x \right )} + \frac {\log {\left (b x \right )}^{2}}{2} + \gamma \log {\left (b x \right )} + i \pi \log {\left (b x \right )} & \text {for}\: \left |{x}\right | < 1 \\b x {{}_{3}F_{3}\left (\begin {matrix} 1, 1, 1 \\ 2, 2, 2 \end {matrix}\middle | {b x} \right )} + i \pi \log {\left (\frac {1}{x} \right )} + \frac {\log {\left (b x \right )}^{2}}{2} + \gamma \log {\left (b x \right )} + i \pi \log {\left (b x \right )} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\b x {{}_{3}F_{3}\left (\begin {matrix} 1, 1, 1 \\ 2, 2, 2 \end {matrix}\middle | {b x} \right )} + i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} - i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} + \frac {\log {\left (b x \right )}^{2}}{2} + \gamma \log {\left (b x \right )} + i \pi \log {\left (b x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(Ei(b*x)/x,x)
Output:
Piecewise((b*x*hyper((1, 1, 1), (2, 2, 2), b*x) + I*pi*log(1/x) - I*pi*log (x) + log(b*x)**2/2 + EulerGamma*log(b*x) + I*pi*log(b*x), (Abs(x) < 1) & (1/Abs(x) < 1)), (b*x*hyper((1, 1, 1), (2, 2, 2), b*x) - I*pi*log(x) + log (b*x)**2/2 + EulerGamma*log(b*x) + I*pi*log(b*x), Abs(x) < 1), (b*x*hyper( (1, 1, 1), (2, 2, 2), b*x) + I*pi*log(1/x) + log(b*x)**2/2 + EulerGamma*lo g(b*x) + I*pi*log(b*x), 1/Abs(x) < 1), (b*x*hyper((1, 1, 1), (2, 2, 2), b* x) + I*pi*meijerg(((), (1, 1)), ((0, 0), ()), x) - I*pi*meijerg(((1, 1), ( )), ((), (0, 0)), x) + log(b*x)**2/2 + EulerGamma*log(b*x) + I*pi*log(b*x) , True))
\[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\int { \frac {{\rm Ei}\left (b x\right )}{x} \,d x } \] Input:
integrate(Ei(b*x)/x,x, algorithm="maxima")
Output:
integrate(Ei(b*x)/x, x)
\[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\int { \frac {{\rm Ei}\left (b x\right )}{x} \,d x } \] Input:
integrate(Ei(b*x)/x,x, algorithm="giac")
Output:
integrate(Ei(b*x)/x, x)
Timed out. \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\int \frac {\mathrm {ei}\left (b\,x\right )}{x} \,d x \] Input:
int(ei(b*x)/x,x)
Output:
int(ei(b*x)/x, x)
\[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x} \, dx=\int \frac {\mathit {ei} \left (b x \right )}{x}d x \] Input:
int(Ei(b*x)/x,x)
Output:
int(ei(b*x)/x,x)