Optimal. Leaf size=37 \[ \frac {\sin (1) \text {Ci}\left (1+2^x\right )}{\log (2)}+\frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2282, 6742, 3299, 3303, 3302} \[ \frac {\sin (1) \text {CosIntegral}\left (2^x+1\right )}{\log (2)}+\frac {\text {Si}\left (2^x\right )}{\log (2)}-\frac {\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sin \left (2^x\right )}{1+2^x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x (1+x)} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (x)}{x}-\frac {\sin (x)}{1+x}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,2^x\right )}{\log (2)}-\frac {\operatorname {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) \operatorname {Subst}\left (\int \frac {\sin (1+x)}{1+x} \, dx,x,2^x\right )}{\log (2)}+\frac {\sin (1) \operatorname {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\text {Ci}\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] time = 0.07, size = 29, normalized size = 0.78 \[ \frac {\sin (1) \text {Ci}\left (1+2^x\right )+\text {Si}\left (2^x\right )-\cos (1) \text {Si}\left (1+2^x\right )}{\log (2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.85, size = 43, normalized size = 1.16 \[ \frac {\operatorname {Ci}\left (2^{x} + 1\right ) \sin \relax (1) + \operatorname {Ci}\left (-2^{x} - 1\right ) \sin \relax (1) - 2 \, \cos \relax (1) \operatorname {Si}\left (2^{x} + 1\right ) + 2 \, \operatorname {Si}\left (2^{x}\right )}{2 \, \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 29, normalized size = 0.78 \[ \frac {\operatorname {Ci}\left (2^{x} + 1\right ) \sin \relax (1) - \cos \relax (1) \operatorname {Si}\left (2^{x} + 1\right ) + \operatorname {Si}\left (2^{x}\right )}{\log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 38, normalized size = 1.03 \[ \frac {\Si \left (2^{x}\right )}{\ln \relax (2)}-\frac {\cos \relax (1) \Si \left (1+2^{x}\right )}{\ln \relax (2)}+\frac {\Ci \left (1+2^{x}\right ) \sin \relax (1)}{\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (2^{x}\right )}{2^{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sin \left (2^x\right )}{2^x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (2^{x} \right )}}{2^{x} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________