3.56.70 \(\int \frac {-2 x+2 (i \pi +\log (\frac {21}{5}))-e^{\frac {2+x}{2}} x (i \pi +\log (\frac {21}{5}))}{2 x (i \pi +\log (\frac {21}{5}))} \, dx\)

Optimal. Leaf size=31 \[ -e^{\frac {1}{4} (4+2 x)}-\frac {x}{i \pi +\log \left (\frac {21}{5}\right )}+\log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.77, number of steps used = 6, number of rules used = 4, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 14, 2194, 43} \begin {gather*} -e^{\frac {x}{2}+1}-\frac {x}{\log \left (\frac {21}{5}\right )+i \pi }+\frac {\left (\log \left (\frac {441}{25}\right )+2 i \pi \right ) \log (x)}{2 \left (\log \left (\frac {21}{5}\right )+i \pi \right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x + 2*(I*Pi + Log[21/5]) - E^((2 + x)/2)*x*(I*Pi + Log[21/5]))/(2*x*(I*Pi + Log[21/5])),x]

[Out]

-E^(1 + x/2) - x/(I*Pi + Log[21/5]) + (((2*I)*Pi + Log[441/25])*Log[x])/(2*(I*Pi + Log[21/5]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{x} \, dx}{2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )}\\ &=\frac {\int \left (-i e^{1+\frac {x}{2}} \left (\pi -i \log \left (\frac {21}{5}\right )\right )+\frac {i \left (2 \pi +2 i x-i \log \left (\frac {441}{25}\right )\right )}{x}\right ) \, dx}{2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )}\\ &=-\left (\frac {1}{2} \int e^{1+\frac {x}{2}} \, dx\right )+\frac {\int \frac {2 \pi +2 i x-i \log \left (\frac {441}{25}\right )}{x} \, dx}{2 \left (\pi -i \log \left (\frac {21}{5}\right )\right )}\\ &=-e^{1+\frac {x}{2}}+\frac {\int \left (2 i+\frac {2 \pi -i \log \left (\frac {441}{25}\right )}{x}\right ) \, dx}{2 \left (\pi -i \log \left (\frac {21}{5}\right )\right )}\\ &=-e^{1+\frac {x}{2}}-\frac {x}{i \pi +\log \left (\frac {21}{5}\right )}+\frac {\left (2 \pi -i \log \left (\frac {441}{25}\right )\right ) \log (x)}{2 \left (\pi -i \log \left (\frac {21}{5}\right )\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 29, normalized size = 0.94 \begin {gather*} -e^{1+\frac {x}{2}}-\frac {x}{i \pi +\log \left (\frac {21}{5}\right )}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + 2*(I*Pi + Log[21/5]) - E^((2 + x)/2)*x*(I*Pi + Log[21/5]))/(2*x*(I*Pi + Log[21/5])),x]

[Out]

-E^(1 + x/2) - x/(I*Pi + Log[21/5]) + Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 37, normalized size = 1.19 \begin {gather*} \frac {{\left (-i \, \pi - \log \left (\frac {21}{5}\right )\right )} e^{\left (\frac {1}{2} \, x + 1\right )} + {\left (i \, \pi + \log \left (\frac {21}{5}\right )\right )} \log \relax (x) - x}{i \, \pi + \log \left (\frac {21}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/(log(21/5)+I*pi),x, algorithm="frica
s")

[Out]

((-I*pi - log(21/5))*e^(1/2*x + 1) + (I*pi + log(21/5))*log(x) - x)/(I*pi + log(21/5))

________________________________________________________________________________________

giac [A]  time = 0.16, size = 42, normalized size = 1.35 \begin {gather*} \frac {-i \, \pi e^{\left (\frac {1}{2} \, x + 1\right )} - e^{\left (\frac {1}{2} \, x + 1\right )} \log \left (\frac {21}{5}\right ) + i \, \pi \log \relax (x) + \log \left (\frac {21}{5}\right ) \log \relax (x) - x}{i \, \pi + \log \left (\frac {21}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/(log(21/5)+I*pi),x, algorithm="giac"
)

[Out]

(-I*pi*e^(1/2*x + 1) - e^(1/2*x + 1)*log(21/5) + I*pi*log(x) + log(21/5)*log(x) - x)/(I*pi + log(21/5))

________________________________________________________________________________________

maple [A]  time = 0.22, size = 45, normalized size = 1.45

method	result	size
norman	\(\frac {\left (i \pi +\ln \relax (5)-\ln \left (21\right )\right ) x}{\pi ^{2}+\ln \left (21\right )^{2}-2 \ln \left (21\right ) \ln \relax (5)+\ln \relax (5)^{2}}-{\mathrm e}^{1+\frac {x}{2}}+\ln \relax (x )\)	\(45\)
derivativedivides	\(\frac {2 i \pi \ln \left (\frac {x}{2}\right )-2 i \pi \,{\mathrm e} \expIntegralEi \left (1, -\frac {x}{2}\right )-2 i \pi \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \expIntegralEi \left (1, -\frac {x}{2}\right )\right )-4-2 x -2 \ln \relax (5) \ln \left (\frac {x}{2}\right )+2 \ln \left (21\right ) \ln \left (\frac {x}{2}\right )+2 \,{\mathrm e}^{1+\frac {x}{2}} \ln \relax (5)-2 \,{\mathrm e}^{1+\frac {x}{2}} \ln \left (21\right )}{2 \ln \left (\frac {21}{5}\right )+2 i \pi }\)	\(92\)
default	\(\frac {2 i \pi \ln \left (\frac {x}{2}\right )-2 i \pi \,{\mathrm e} \expIntegralEi \left (1, -\frac {x}{2}\right )-2 i \pi \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \expIntegralEi \left (1, -\frac {x}{2}\right )\right )-4-2 x -2 \ln \relax (5) \ln \left (\frac {x}{2}\right )+2 \ln \left (21\right ) \ln \left (\frac {x}{2}\right )+2 \,{\mathrm e}^{1+\frac {x}{2}} \ln \relax (5)-2 \,{\mathrm e}^{1+\frac {x}{2}} \ln \left (21\right )}{2 \ln \left (\frac {21}{5}\right )+2 i \pi }\)	\(92\)
risch	\(-\frac {x}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }-\frac {\ln \relax (x ) \ln \relax (5)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }+\frac {\ln \relax (x ) \ln \relax (7)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }+\frac {\ln \relax (x ) \ln \relax (3)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }+\frac {i \ln \relax (x ) \pi }{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }-\frac {i {\mathrm e}^{1+\frac {x}{2}} \pi }{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }+\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \relax (5)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }-\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \relax (7)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }-\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \relax (3)}{\ln \relax (3)+\ln \relax (7)-\ln \relax (5)+i \pi }\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-x*(ln(21/5)+I*Pi)*exp(1+1/2*x)+2*ln(21/5)+2*I*Pi-2*x)/x/(ln(21/5)+I*Pi),x,method=_RETURNVERBOSE)

[Out]

(I*Pi+ln(5)-ln(21))/(Pi^2+ln(21)^2-2*ln(21)*ln(5)+ln(5)^2)*x-exp(1+1/2*x)+ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 42, normalized size = 1.35 \begin {gather*} \frac {-i \, \pi e^{\left (\frac {1}{2} \, x + 1\right )} - e^{\left (\frac {1}{2} \, x + 1\right )} \log \left (\frac {21}{5}\right ) + i \, \pi \log \relax (x) + \log \left (\frac {21}{5}\right ) \log \relax (x) - x}{i \, \pi + \log \left (\frac {21}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/(log(21/5)+I*pi),x, algorithm="maxim
a")

[Out]

(-I*pi*e^(1/2*x + 1) - e^(1/2*x + 1)*log(21/5) + I*pi*log(x) + log(21/5)*log(x) - x)/(I*pi + log(21/5))

________________________________________________________________________________________

mupad [B]  time = 3.66, size = 51, normalized size = 1.65 \begin {gather*} \ln \relax (x)-\frac {{\mathrm {e}}^{\frac {x}{2}+1}\,\left (2\,\Pi ^2+2\,{\ln \left (\frac {21}{5}\right )}^2\right )-x\,\left (-2\,\ln \left (\frac {21}{5}\right )+\Pi \,2{}\mathrm {i}\right )}{2\,\Pi ^2+2\,{\ln \left (\frac {21}{5}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((Pi*1i - x + log(21/5) - (x*exp(x/2 + 1)*(Pi*1i + log(21/5)))/2)/(x*(Pi*1i + log(21/5))),x)

[Out]

log(x) - (exp(x/2 + 1)*(2*Pi^2 + 2*log(21/5)^2) - x*(Pi*2i - 2*log(21/5)))/(2*Pi^2 + 2*log(21/5)^2)

________________________________________________________________________________________

sympy [A]  time = 0.39, size = 32, normalized size = 1.03 \begin {gather*} \frac {x + \left (- \log {\left (21 \right )} + \log {\relax (5 )} - i \pi \right ) \log {\relax (x )}}{- \log {\left (21 \right )} + \log {\relax (5 )} - i \pi } - e e^{\frac {x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*(ln(21/5)+I*pi)*exp(1+1/2*x)+2*ln(21/5)+2*I*pi-2*x)/x/(ln(21/5)+I*pi),x)

[Out]

(x + (-log(21) + log(5) - I*pi)*log(x))/(-log(21) + log(5) - I*pi) - E*exp(x/2)

________________________________________________________________________________________