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} $$\begin{array}{c}{e}^x-\frac{{\left(\frac{3}{16}\right)}^{\frac{1}{x}}{e}^{\frac{i\pi }{x}}}{x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (e^x+\frac{{\left(\frac{3}{16}\right)}^{\frac{1}{x}}{e}^{\frac{i\pi }{x}}(i\pi +x-\log \left(\frac{16}{3}\right))}{x^3})dx\\ & =\int e^{x}dx+\int \frac{{\left(\frac{3}{16}\right)}^{\frac{1}{x}}{e}^{\frac{i\pi }{x}}(i\pi +x-\log \left(\frac{16}{3}\right))}{x^3}dx\\ & =e^x+\int \frac{e^{\frac{i\pi -\log \left(\frac{16}{3}\right)}{x}}(i\pi +x-\log \left(\frac{16}{3}\right))}{x^3}dx\\ & =e^x-\frac{{\left(\frac{3}{16}\right)}^{\frac{1}{x}}{e}^{\frac{i\pi }{x}}}{x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.05, size = 27, normalized size = 1.00 $$\begin{array}{c}{e}^x-\frac{e^{\frac{i\pi -\log \left(\frac{16}{3}\right)}{x}}}{x}\end{array}$$

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 $$\begin{array}{c}\frac{x{e}^x-e^{(\frac{i\pi }{x}+\frac{\log \left(\frac{3}{16}\right)}{x})}}{x}\end{array}$$

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 $$\begin{array}{c}\frac{x{e}^x-e^{(\frac{i\pi }{x}+\frac{\log \left(\frac{3}{16}\right)}{x})}}{x}\end{array}$$

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

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	$-\frac{1-{\mathrm{e}}^{-\frac{-\ln \left(\frac{3}{16}\right)-i\pi }{x}}}{-\ln \left(\frac{3}{16}\right)-i\pi }-\frac{(\ln \left(\frac{3}{16}\right)+i\pi )(1-\frac{(\frac{-2\ln \left(\frac{3}{16}\right)-2i\pi }{x}+2){\mathrm{e}}^{-\frac{-\ln \left(\frac{3}{16}\right)-i\pi }{x}}}{2})}{{(-\ln \left(\frac{3}{16}\right)-i\pi )}^2}-1+{\mathrm{e}}^x$	$92$

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 $$\begin{array}{c}{e}^x+\int \frac{(i\pi +x+\log (3)-4\log (2))e^{(\frac{i\pi }{x}+\frac{\log (3)}{x}-\frac{4\log (2)}{x})}}{x^3}dx\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^x-\frac{3^{1/x}{\mathrm{e}}^{\frac{\mathrm{\Pi }1\mathrm{i}}{x}}}{2^{4/x}x}\end{array}$$

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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