3.33.78 \(\int \frac {e^x x^3+e^{\frac {i \pi -\log (\frac {16}{3})}{x}} (i \pi +x-\log (\frac {16}{3}))}{x^3} \, dx\)
Optimal. Leaf size=27 \[ e^x-\frac {e^{\frac {i \pi -\log \left (\frac {16}{3}\right )}{x}}}{x} \]
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Rubi [A] time = 0.21, antiderivative size = 26, normalized size of antiderivative = 0.96,
number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used =
{14, 2194, 2287, 2288} \begin {gather*} e^x-\frac {\left (\frac {3}{16}\right )^{\frac {1}{x}} e^{\frac {i \pi }{x}}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(E^x*x^3 + E^((I*Pi - Log[16/3])/x)*(I*Pi + x - Log[16/3]))/x^3,x]
[Out]
E^x - ((3/16)^x^(-1)*E^((I*Pi)/x))/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 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]
Rule 2287
Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]
Rule 2288
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {\left (\frac {3}{16}\right )^{\frac {1}{x}} e^{\frac {i \pi }{x}} \left (i \pi +x-\log \left (\frac {16}{3}\right )\right )}{x^3}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {\left (\frac {3}{16}\right )^{\frac {1}{x}} e^{\frac {i \pi }{x}} \left (i \pi +x-\log \left (\frac {16}{3}\right )\right )}{x^3} \, dx\\ &=e^x+\int \frac {e^{\frac {i \pi -\log \left (\frac {16}{3}\right )}{x}} \left (i \pi +x-\log \left (\frac {16}{3}\right )\right )}{x^3} \, dx\\ &=e^x-\frac {\left (\frac {3}{16}\right )^{\frac {1}{x}} e^{\frac {i \pi }{x}}}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 1.00 \begin {gather*} e^x-\frac {e^{\frac {i \pi -\log \left (\frac {16}{3}\right )}{x}}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^x*x^3 + E^((I*Pi - Log[16/3])/x)*(I*Pi + x - Log[16/3]))/x^3,x]
[Out]
E^x - E^((I*Pi - Log[16/3])/x)/x
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fricas [A] time = 0.58, size = 25, normalized size = 0.93 \begin {gather*} \frac {x e^{x} - e^{\left (\frac {i \, \pi }{x} + \frac {\log \left (\frac {3}{16}\right )}{x}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((log(3/16)+I*pi+x)*exp((log(3/16)+I*pi)/x)+exp(x)*x^3)/x^3,x, algorithm="fricas")
[Out]
(x*e^x - e^(I*pi/x + log(3/16)/x))/x
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giac [A] time = 0.46, size = 25, normalized size = 0.93 \begin {gather*} \frac {x e^{x} - e^{\left (\frac {i \, \pi }{x} + \frac {\log \left (\frac {3}{16}\right )}{x}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((log(3/16)+I*pi+x)*exp((log(3/16)+I*pi)/x)+exp(x)*x^3)/x^3,x, algorithm="giac")
[Out]
(x*e^x - e^(I*pi/x + log(3/16)/x))/x
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maple [A] time = 0.12, size = 27, normalized size = 1.00
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method |
result |
size |
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norman |
\(\frac {{\mathrm e}^{x} x^{2}-{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}} x}{x^{2}}\) |
\(27\) |
risch |
\({\mathrm e}^{x}-\frac {3^{\frac {1}{x}} \left (\frac {1}{16}\right )^{\frac {1}{x}} {\mathrm e}^{\frac {i \pi }{x}}}{x}\) |
\(27\) |
meijerg |
\(-\frac {1-{\mathrm e}^{-\frac {-\ln \left (\frac {3}{16}\right )-i \pi }{x}}}{-\ln \left (\frac {3}{16}\right )-i \pi }-\frac {\left (\ln \left (\frac {3}{16}\right )+i \pi \right ) \left (1-\frac {\left (\frac {-2 \ln \left (\frac {3}{16}\right )-2 i \pi }{x}+2\right ) {\mathrm e}^{-\frac {-\ln \left (\frac {3}{16}\right )-i \pi }{x}}}{2}\right )}{\left (-\ln \left (\frac {3}{16}\right )-i \pi \right )^{2}}-1+{\mathrm e}^{x}\) |
\(92\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((ln(3/16)+I*Pi+x)*exp((ln(3/16)+I*Pi)/x)+exp(x)*x^3)/x^3,x,method=_RETURNVERBOSE)
[Out]
(exp(x)*x^2-exp((ln(3/16)+I*Pi)/x)*x)/x^2
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{x} + \int \frac {{\left (i \, \pi + x + \log \relax (3) - 4 \, \log \relax (2)\right )} e^{\left (\frac {i \, \pi }{x} + \frac {\log \relax (3)}{x} - \frac {4 \, \log \relax (2)}{x}\right )}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((log(3/16)+I*pi+x)*exp((log(3/16)+I*pi)/x)+exp(x)*x^3)/x^3,x, algorithm="maxima")
[Out]
e^x + integrate((I*pi + x + log(3) - 4*log(2))*e^(I*pi/x + log(3)/x - 4*log(2)/x)/x^3, x)
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mupad [B] time = 2.04, size = 30, normalized size = 1.11 \begin {gather*} {\mathrm {e}}^x-\frac {3^{1/x}\,{\mathrm {e}}^{\frac {\Pi \,1{}\mathrm {i}}{x}}}{2^{4/x}\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3*exp(x) + exp((Pi*1i + log(3/16))/x)*(Pi*1i + x + log(3/16)))/x^3,x)
[Out]
exp(x) - (3^(1/x)*exp((Pi*1i)/x))/(2^(4/x)*x)
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sympy [B] time = 91.75, size = 2186, normalized size = 80.96 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((ln(3/16)+I*pi+x)*exp((ln(3/16)+I*pi)/x)+exp(x)*x**3)/x**3,x)
[Out]
-4*x*log(2)/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*
pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(
3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*lo
g(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) + x*log(3)/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) -
8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)
*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)
/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) + I*pi*x/(-pi**2*x*exp(
4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*ex
p(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*
*2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp
(-I*pi/x)*log(3)) - x*exp(4*log(2)/x)*log(3)/(-pi**2*x*exp(4*log(2)/x) - 8*x*exp(4*log(2)/x)*log(2)*log(3) + x
*exp(4*log(2)/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*log(2) + 2*I*pi*x*exp(4
*log(2)/x)*log(3)) + 4*x*exp(4*log(2)/x)*log(2)/(-pi**2*x*exp(4*log(2)/x) - 8*x*exp(4*log(2)/x)*log(2)*log(3)
+ x*exp(4*log(2)/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*log(2) + 2*I*pi*x*ex
p(4*log(2)/x)*log(3)) - I*pi*x*exp(4*log(2)/x)/(-pi**2*x*exp(4*log(2)/x) - 8*x*exp(4*log(2)/x)*log(2)*log(3) +
x*exp(4*log(2)/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*log(2) + 2*I*pi*x*exp
(4*log(2)/x)*log(3)) + exp(x) - 16*log(2)**2/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4
*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2
+ 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I
*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) - log(3)**2/(-pi**2*x*exp(4*log(2
)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log
(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*
I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/
x)*log(3)) + 8*log(2)*log(3)/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-
log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*lo
g(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) +
2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) + pi**2/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*
exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x
)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/
x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) - 2*I*pi*
log(3)/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) - 8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)
*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)
*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/
x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) + 8*I*pi*log(2)/(-pi**2*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x) -
8*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)*log(3) + x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)
*log(3)**2 + 16*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2)**2 - 8*I*pi*x*exp(4*log(2)/x)*exp(-log(3)
/x)*exp(-I*pi/x)*log(2) + 2*I*pi*x*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(3)) + 1/(-exp(4*log(2)/x)*e
xp(-log(3)/x)*exp(-I*pi/x)*log(3) + 4*exp(4*log(2)/x)*exp(-log(3)/x)*exp(-I*pi/x)*log(2) - I*pi*exp(4*log(2)/x
)*exp(-log(3)/x)*exp(-I*pi/x)) - exp(4*log(2)/x)/(-exp(4*log(2)/x)*log(3) + 4*exp(4*log(2)/x)*log(2) - I*pi*ex
p(4*log(2)/x))
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