3.33.78 \(\int \frac {e^x x^3+e^{\frac {i \pi -\log (\frac {16}{3})}{x}} (i \pi +x-\log (\frac {16}{3}))}{x^3} \, dx\) [3278]

Optimal. Leaf size=27 \[ e^x-\frac {e^{\frac {i \pi -\log \left (\frac {16}{3}\right )}{x}}}{x} \]

[Out]

exp(x)-exp((ln(3/16)+I*Pi)/x)/x

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Rubi [A]
time = 0.19, 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, 2225, 2325, 2326} \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 2225

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 2325

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 2326

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 = 0.04, 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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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (20 ) = 40\).
time = 0.62, size = 192, normalized size = 7.11

method	result	size
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	\(-\frac {1-{\mathrm e}^{-\frac {-\ln \left (\frac {3}{16}\right )-i \pi }{x}}}{-\ln \left (\frac {3}{16}\right )-i \pi }-\frac {\left (\ln \left (\frac {3}{16}\right )+i \pi \right ) \left (1-\frac {\left (2+\frac {-2 \ln \left (\frac {3}{16}\right )-2 i \pi }{x}\right ) {\mathrm e}^{-\frac {-\ln \left (\frac {3}{16}\right )-i \pi }{x}}}{2}\right )}{\left (-\ln \left (\frac {3}{16}\right )-i \pi \right )^{2}}-1+{\mathrm e}^{x}\)	\(92\)
default	\({\mathrm e}^{x}-\frac {{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}}}{\ln \left (\frac {3}{16}\right )+i \pi }+i \pi \left (-\frac {i {\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}} \pi }{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2} x}-\frac {{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}} \ln \left (\frac {3}{16}\right )}{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2} x}+\frac {{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}}}{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2}}\right )+\ln \left (\frac {3}{16}\right ) \left (-\frac {i {\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}} \pi }{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2} x}-\frac {{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}} \ln \left (\frac {3}{16}\right )}{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2} x}+\frac {{\mathrm e}^{\frac {\ln \left (\frac {3}{16}\right )+i \pi }{x}}}{\left (\ln \left (\frac {3}{16}\right )+i \pi \right )^{2}}\right )\)	\(192\)

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)-exp((ln(3/16)+I*Pi)/x)/(ln(3/16)+I*Pi)+I*Pi*(-I/(ln(3/16)+I*Pi)^2*exp((ln(3/16)+I*Pi)/x)/x*Pi-1/(ln(3/1
6)+I*Pi)^2*exp((ln(3/16)+I*Pi)/x)/x*ln(3/16)+1/(ln(3/16)+I*Pi)^2*exp((ln(3/16)+I*Pi)/x))+ln(3/16)*(-I/(ln(3/16
)+I*Pi)^2*exp((ln(3/16)+I*Pi)/x)/x*Pi-1/(ln(3/16)+I*Pi)^2*exp((ln(3/16)+I*Pi)/x)/x*ln(3/16)+1/(ln(3/16)+I*Pi)^
2*exp((ln(3/16)+I*Pi)/x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 0.39, 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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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (15) = 30\).
time = 48.01, size = 1321, normalized size = 48.93 \begin {gather*} \text {Too large to display} \end {gather*}

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*exp(log(3)/x)*exp(I*pi/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*exp(4*log(2)
/x)*log(3)) + x*exp(log(3)/x)*exp(I*pi/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*e
xp(4*log(2)/x)*log(3)) + I*pi*x*exp(log(3)/x)*exp(I*pi/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)) - x*log(3)/(-pi**2*x - 8*x*log(2)*log(3) + x*log(3)**2 + 16*x*log(2)**2 - 8*I*
pi*x*log(2) + 2*I*pi*x*log(3)) + 4*x*log(2)/(-pi**2*x - 8*x*log(2)*log(3) + x*log(3)**2 + 16*x*log(2)**2 - 8*I
*pi*x*log(2) + 2*I*pi*x*log(3)) - I*pi*x/(-pi**2*x - 8*x*log(2)*log(3) + x*log(3)**2 + 16*x*log(2)**2 - 8*I*pi
*x*log(2) + 2*I*pi*x*log(3)) + exp(x) - 16*exp(log(3)/x)*exp(I*pi/x)*log(2)**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*exp(4*log(2)/x)*log(3)) - exp(log(3)/x)*exp(I*pi/x)*log(3)**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*exp(4*log(2)/x)*log(3)) + 8*exp(log(3)/x)*exp(I*pi/x)*log(2)*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)) + pi**2*exp(log(3)/x)*exp(I
*pi/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)) - 2*I*pi*exp(log(3)
/x)*exp(I*pi/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)) + 8
*I*pi*exp(log(3)/x)*exp(I*pi/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*exp(4*log(2
)/x)*log(3)) + exp(log(3)/x)*exp(I*pi/x)/(-exp(4*log(2)/x)*log(3) + 4*exp(4*log(2)/x)*log(2) - I*pi*exp(4*log(
2)/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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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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