\(\int \frac {\sin (2^x)}{1+2^x} \, dx\) [922]

Optimal result

Integrand size = 12, antiderivative size = 37 \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\frac {\operatorname {CosIntegral}\left (1+2^x\right ) \sin (1)}{\log (2)}+\frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]

[Out]

Si(2^x)/ln(2)-cos(1)*Si(1+2^x)/ln(2)+Ci(1+2^x)*sin(1)/ln(2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2320, 6874, 3380, 3384, 3383} \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\frac {\sin (1) \operatorname {CosIntegral}\left (1+2^x\right )}{\log (2)}+\frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]

[In]

Int[Sin[2^x]/(1 + 2^x),x]

[Out]

(CosIntegral[1 + 2^x]*Sin[1])/Log[2] + SinIntegral[2^x]/Log[2] - (Cos[1]*SinIntegral[1 + 2^x])/Log[2]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sin (x)}{x (1+x)} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{x}-\frac {\sin (x)}{1+x}\right ) \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,2^x\right )}{\log (2)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{1+x} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Subst}\left (\int \frac {\sin (1+x)}{1+x} \, dx,x,2^x\right )}{\log (2)}+\frac {\sin (1) \text {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\operatorname {CosIntegral}\left (1+2^x\right ) \sin (1)}{\log (2)}+\frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\frac {\operatorname {CosIntegral}\left (1+2^x\right ) \sin (1)+\text {Si}\left (2^x\right )-\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]

[In]

Integrate[Sin[2^x]/(1 + 2^x),x]

[Out]

(CosIntegral[1 + 2^x]*Sin[1] + SinIntegral[2^x] - Cos[1]*SinIntegral[1 + 2^x])/Log[2]

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {-\operatorname {Si}\left (1+2^{x}\right ) \cos \left (1\right )+\operatorname {Ci}\left (1+2^{x}\right ) \sin \left (1\right )+\operatorname {Si}\left (2^{x}\right )}{\ln \left (2\right )}\)	\(30\)
default	\(\frac {-\operatorname {Si}\left (1+2^{x}\right ) \cos \left (1\right )+\operatorname {Ci}\left (1+2^{x}\right ) \sin \left (1\right )+\operatorname {Si}\left (2^{x}\right )}{\ln \left (2\right )}\)	\(30\)
risch	\(-\frac {i \operatorname {Ei}_{1}\left (-i 2^{x}-i\right ) {\mathrm e}^{-i}}{2 \ln \left (2\right )}+\frac {i \operatorname {Ei}_{1}\left (i 2^{x}+i\right ) {\mathrm e}^{i}}{2 \ln \left (2\right )}+\frac {i \operatorname {Ei}_{1}\left (-i 2^{x}\right )}{2 \ln \left (2\right )}-\frac {i \operatorname {Ei}_{1}\left (i 2^{x}\right )}{2 \ln \left (2\right )}\)	\(74\)

[In]

int(sin(2^x)/(1+2^x),x,method=_RETURNVERBOSE)

[Out]

1/ln(2)*(-Si(1+2^x)*cos(1)+Ci(1+2^x)*sin(1)+Si(2^x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\frac {\operatorname {Ci}\left (2^{x} + 1\right ) \sin \left (1\right ) - \cos \left (1\right ) \operatorname {Si}\left (2^{x} + 1\right ) + \operatorname {Si}\left (2^{x}\right )}{\log \left (2\right )} \]

[In]

integrate(sin(2^x)/(1+2^x),x, algorithm="fricas")

[Out]

(cos_integral(2^x + 1)*sin(1) - cos(1)*sin_integral(2^x + 1) + sin_integral(2^x))/log(2)

Sympy [F]

\[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\int \frac {\sin {\left (2^{x} \right )}}{2^{x} + 1}\, dx \]

[In]

integrate(sin(2**x)/(1+2**x),x)

[Out]

Integral(sin(2**x)/(2**x + 1), x)

Maxima [F]

\[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\int { \frac {\sin \left (2^{x}\right )}{2^{x} + 1} \,d x } \]

[In]

integrate(sin(2^x)/(1+2^x),x, algorithm="maxima")

[Out]

integrate(sin(2^x)/(2^x + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\frac {\operatorname {Ci}\left (2^{x} + 1\right ) \sin \left (1\right ) - \cos \left (1\right ) \operatorname {Si}\left (2^{x} + 1\right ) + \operatorname {Si}\left (2^{x}\right )}{\log \left (2\right )} \]

[In]

integrate(sin(2^x)/(1+2^x),x, algorithm="giac")

[Out]

(cos_integral(2^x + 1)*sin(1) - cos(1)*sin_integral(2^x + 1) + sin_integral(2^x))/log(2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx=\int \frac {\sin \left (2^x\right )}{2^x+1} \,d x \]

[In]

int(sin(2^x)/(2^x + 1),x)

[Out]

int(sin(2^x)/(2^x + 1), x)