Integrand size = 16, antiderivative size = 79 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {48 \cos \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}-\frac {2 \cos ^3\left (\frac {x}{3}\right )}{5 \sqrt {e^x}}+\frac {32 \sin \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}+\frac {4 \cos ^2\left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )}{5 \sqrt {e^x}} \] Output:
-48/65*cos(1/3*x)/exp(x)^(1/2)-2/5*cos(1/3*x)^3/exp(x)^(1/2)+32/65*sin(1/3 *x)/exp(x)^(1/2)+4/5*cos(1/3*x)^2*sin(1/3*x)/exp(x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {-135 \cos \left (\frac {x}{3}\right )-13 \cos (x)+90 \sin \left (\frac {x}{3}\right )+26 \sin (x)}{130 \sqrt {e^x}} \] Input:
Integrate[Cos[x/3]^3/Sqrt[E^x],x]
Output:
(-135*Cos[x/3] - 13*Cos[x] + 90*Sin[x/3] + 26*Sin[x])/(130*Sqrt[E^x])
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2717, 4935, 4933}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx\) |
\(\Big \downarrow \) 2717 |
\(\displaystyle \frac {e^{x/2} \int e^{-x/2} \cos ^3\left (\frac {x}{3}\right )dx}{\sqrt {e^x}}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {e^{x/2} \left (\frac {8}{15} \int e^{-x/2} \cos \left (\frac {x}{3}\right )dx-\frac {2}{5} e^{-x/2} \cos ^3\left (\frac {x}{3}\right )+\frac {4}{5} e^{-x/2} \sin \left (\frac {x}{3}\right ) \cos ^2\left (\frac {x}{3}\right )\right )}{\sqrt {e^x}}\) |
\(\Big \downarrow \) 4933 |
\(\displaystyle \frac {e^{x/2} \left (-\frac {2}{5} e^{-x/2} \cos ^3\left (\frac {x}{3}\right )+\frac {4}{5} e^{-x/2} \sin \left (\frac {x}{3}\right ) \cos ^2\left (\frac {x}{3}\right )+\frac {8}{15} \left (\frac {12}{13} e^{-x/2} \sin \left (\frac {x}{3}\right )-\frac {18}{13} e^{-x/2} \cos \left (\frac {x}{3}\right )\right )\right )}{\sqrt {e^x}}\) |
Input:
Int[Cos[x/3]^3/Sqrt[E^x],x]
Output:
(E^(x/2)*((-2*Cos[x/3]^3)/(5*E^(x/2)) + (4*Cos[x/3]^2*Sin[x/3])/(5*E^(x/2) ) + (8*((-18*Cos[x/3])/(13*E^(x/2)) + (12*Sin[x/3])/(13*E^(x/2))))/15))/Sq rt[E^x]
Int[(u_.)*((a_.)*(F_)^(v_))^(n_), x_Symbol] :> Simp[(a*F^v)^n/F^(n*v) Int [u*F^(n*v), x], x] /; FreeQ[{F, a, n}, x] && !IntegerQ[n]
Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x ] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] /; F reeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbo l] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Lo g[F]^2)), x] + (Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1) /(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + Simp[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^ 2*Log[F]^2) Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x]) /; FreeQ[{F , a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
Time = 0.64 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.35
method | result | size |
parallelrisch | \(\frac {-13 \cos \left (x \right )-135 \cos \left (\frac {x}{3}\right )+90 \sin \left (\frac {x}{3}\right )+26 \sin \left (x \right )}{130 \sqrt {{\mathrm e}^{x}}}\) | \(28\) |
default | \(-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (x \right )}{10}+\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (x \right )}{5}-\frac {27 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {x}{3}\right )}{26}+\frac {9 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {x}{3}\right )}{13}\) | \(38\) |
risch | \(\frac {\left (-\frac {1}{1300}-\frac {i}{650}\right ) \left (-52 i {\mathrm e}^{-i x}+65 \,{\mathrm e}^{i x}-39 \,{\mathrm e}^{-i x}+\left (270-540 i\right ) \cos \left (\frac {x}{3}\right )+\left (-180+360 i\right ) \sin \left (\frac {x}{3}\right )\right )}{\sqrt {{\mathrm e}^{x}}}\) | \(48\) |
orering | \(-\frac {74 \cos \left (\frac {x}{3}\right )^{3}}{65 \sqrt {{\mathrm e}^{x}}}+\frac {84 \cos \left (\frac {x}{3}\right )^{2} \sin \left (\frac {x}{3}\right )}{65 \sqrt {{\mathrm e}^{x}}}-\frac {48 \cos \left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )^{2}}{65 \sqrt {{\mathrm e}^{x}}}+\frac {32 \sin \left (\frac {x}{3}\right )^{3}}{65 \sqrt {{\mathrm e}^{x}}}\) | \(58\) |
Input:
int(cos(1/3*x)^3/exp(x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/130*(-13*cos(x)-135*cos(1/3*x)+90*sin(1/3*x)+26*sin(x))/exp(x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {4}{65} \, {\left (13 \, \cos \left (\frac {1}{3} \, x\right )^{2} + 8\right )} e^{\left (-\frac {1}{2} \, x\right )} \sin \left (\frac {1}{3} \, x\right ) - \frac {2}{65} \, {\left (13 \, \cos \left (\frac {1}{3} \, x\right )^{3} + 24 \, \cos \left (\frac {1}{3} \, x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \] Input:
integrate(cos(1/3*x)^3/exp(x)^(1/2),x, algorithm="fricas")
Output:
4/65*(13*cos(1/3*x)^2 + 8)*e^(-1/2*x)*sin(1/3*x) - 2/65*(13*cos(1/3*x)^3 + 24*cos(1/3*x))*e^(-1/2*x)
Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {32 \sin ^{3}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} - \frac {48 \sin ^{2}{\left (\frac {x}{3} \right )} \cos {\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} + \frac {84 \sin {\left (\frac {x}{3} \right )} \cos ^{2}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} - \frac {74 \cos ^{3}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} \] Input:
integrate(cos(1/3*x)**3/exp(x)**(1/2),x)
Output:
32*sin(x/3)**3/(65*sqrt(exp(x))) - 48*sin(x/3)**2*cos(x/3)/(65*sqrt(exp(x) )) + 84*sin(x/3)*cos(x/3)**2/(65*sqrt(exp(x))) - 74*cos(x/3)**3/(65*sqrt(e xp(x)))
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.34 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {1}{130} \, {\left (135 \, \cos \left (\frac {1}{3} \, x\right ) + 13 \, \cos \left (x\right ) - 90 \, \sin \left (\frac {1}{3} \, x\right ) - 26 \, \sin \left (x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \] Input:
integrate(cos(1/3*x)^3/exp(x)^(1/2),x, algorithm="maxima")
Output:
-1/130*(135*cos(1/3*x) + 13*cos(x) - 90*sin(1/3*x) - 26*sin(x))*e^(-1/2*x)
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.42 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {9}{26} \, {\left (3 \, \cos \left (\frac {1}{3} \, x\right ) - 2 \, \sin \left (\frac {1}{3} \, x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} - \frac {1}{10} \, {\left (\cos \left (x\right ) - 2 \, \sin \left (x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \] Input:
integrate(cos(1/3*x)^3/exp(x)^(1/2),x, algorithm="giac")
Output:
-9/26*(3*cos(1/3*x) - 2*sin(1/3*x))*e^(-1/2*x) - 1/10*(cos(x) - 2*sin(x))* e^(-1/2*x)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {8\,{\cos \left (\frac {x}{3}\right )}^3}{5}-\frac {16\,\sin \left (\frac {x}{3}\right )\,{\cos \left (\frac {x}{3}\right )}^2}{5}+\frac {192\,\cos \left (\frac {x}{3}\right )}{65}-\frac {128\,\sin \left (\frac {x}{3}\right )}{65}\right )}{4} \] Input:
int(cos(x/3)^3/exp(x)^(1/2),x)
Output:
-(exp(-x/2)*((192*cos(x/3))/65 - (128*sin(x/3))/65 - (16*cos(x/3)^2*sin(x/ 3))/5 + (8*cos(x/3)^3)/5))/4
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {\frac {2 \cos \left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )^{2}}{5}-\frac {74 \cos \left (\frac {x}{3}\right )}{65}-\frac {4 \sin \left (\frac {x}{3}\right )^{3}}{5}+\frac {84 \sin \left (\frac {x}{3}\right )}{65}}{e^{\frac {x}{2}}} \] Input:
int(cos(1/3*x)^3/exp(x)^(1/2),x)
Output:
(2*(13*cos(x/3)*sin(x/3)**2 - 37*cos(x/3) - 26*sin(x/3)**3 + 42*sin(x/3))) /(65*e**(x/2))