Integrand size = 10, antiderivative size = 82 \[ \int e^{m x} \sin ^3(x) \, dx=-\frac {6 e^{m x} \cos (x)}{9+10 m^2+m^4}+\frac {6 e^{m x} m \sin (x)}{9+10 m^2+m^4}-\frac {3 e^{m x} \cos (x) \sin ^2(x)}{9+m^2}+\frac {e^{m x} m \sin ^3(x)}{9+m^2} \] Output:
-6*exp(m*x)*cos(x)/(m^4+10*m^2+9)+6*exp(m*x)*m*sin(x)/(m^4+10*m^2+9)-3*exp (m*x)*cos(x)*sin(x)^2/(m^2+9)+exp(m*x)*m*sin(x)^3/(m^2+9)
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int e^{m x} \sin ^3(x) \, dx=\frac {e^{m x} \left (-3 \left (9+m^2\right ) \cos (x)+3 \left (1+m^2\right ) \cos (3 x)-2 m \left (-13-m^2+\left (1+m^2\right ) \cos (2 x)\right ) \sin (x)\right )}{4 \left (9+10 m^2+m^4\right )} \] Input:
Integrate[E^(m*x)*Sin[x]^3,x]
Output:
(E^(m*x)*(-3*(9 + m^2)*Cos[x] + 3*(1 + m^2)*Cos[3*x] - 2*m*(-13 - m^2 + (1 + m^2)*Cos[2*x])*Sin[x]))/(4*(9 + 10*m^2 + m^4))
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4934, 4932}
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.
\(\displaystyle \int e^{m x} \sin ^3(x) \, dx\) |
\(\Big \downarrow \) 4934 |
\(\displaystyle \frac {6 \int e^{m x} \sin (x)dx}{m^2+9}+\frac {m e^{m x} \sin ^3(x)}{m^2+9}-\frac {3 e^{m x} \sin ^2(x) \cos (x)}{m^2+9}\) |
\(\Big \downarrow \) 4932 |
\(\displaystyle \frac {m e^{m x} \sin ^3(x)}{m^2+9}-\frac {3 e^{m x} \sin ^2(x) \cos (x)}{m^2+9}+\frac {6 \left (\frac {m e^{m x} \sin (x)}{m^2+1}-\frac {e^{m x} \cos (x)}{m^2+1}\right )}{m^2+9}\) |
Input:
Int[E^(m*x)*Sin[x]^3,x]
Output:
(-3*E^(m*x)*Cos[x]*Sin[x]^2)/(9 + m^2) + (E^(m*x)*m*Sin[x]^3)/(9 + m^2) + (6*(-((E^(m*x)*Cos[x])/(1 + m^2)) + (E^(m*x)*m*Sin[x])/(1 + m^2)))/(9 + m^ 2)
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x ] - Simp[e*F^(c*(a + b*x))*(Cos[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[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbo l] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Sin[d + e*x]^n/(e^2*n^2 + b^2*c^2*Lo g[F]^2)), x] + (-Simp[e*n*F^(c*(a + b*x))*Cos[d + e*x]*(Sin[d + e*x]^(n - 1 )/(e^2*n^2 + b^2*c^2*Log[F]^2)), x] + Simp[(n*(n - 1)*e^2)/(e^2*n^2 + b^2*c ^2*Log[F]^2) Int[F^(c*(a + b*x))*Sin[d + e*x]^(n - 2), x], x]) /; FreeQ[{ F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {i {\mathrm e}^{\left (3 i+m \right ) x}}{24 i+8 m}-\frac {3 i {\mathrm e}^{x \left (m +i\right )}}{8 \left (m +i\right )}+\frac {3 i {\mathrm e}^{x \left (-i+m \right )}}{8 \left (-i+m \right )}-\frac {i {\mathrm e}^{x \left (m -3 i\right )}}{8 \left (m -3 i\right )}\) | \(66\) |
default | \(-\frac {3 \,{\mathrm e}^{m x} \cos \left (x \right )}{4 \left (m^{2}+1\right )}+\frac {3 m \,{\mathrm e}^{m x} \sin \left (x \right )}{4 \left (m^{2}+1\right )}+\frac {3 \,{\mathrm e}^{m x} \cos \left (3 x \right )}{4 \left (m^{2}+9\right )}-\frac {m \,{\mathrm e}^{m x} \sin \left (3 x \right )}{4 \left (m^{2}+9\right )}\) | \(68\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{m x} \left (-3+3 \tan \left (\frac {x}{2}\right )^{6}+6 m \tan \left (\frac {x}{2}\right )^{5}+6 \tan \left (\frac {x}{2}\right )^{4} m^{2}+4 \tan \left (\frac {x}{2}\right )^{3} m^{3}+9 \tan \left (\frac {x}{2}\right )^{4}+16 \tan \left (\frac {x}{2}\right )^{3} m -6 \tan \left (\frac {x}{2}\right )^{2} m^{2}-9 \tan \left (\frac {x}{2}\right )^{2}+6 \tan \left (\frac {x}{2}\right ) m \right )}{\left (m^{2}+9\right ) \left (m^{2}+1\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(115\) |
norman | \(\frac {-\frac {6 \,{\mathrm e}^{m x}}{m^{4}+10 m^{2}+9}+\frac {6 \,{\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )^{6}}{m^{4}+10 m^{2}+9}+\frac {12 m \,{\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )}{m^{4}+10 m^{2}+9}+\frac {12 m \,{\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )^{5}}{m^{4}+10 m^{2}+9}-\frac {6 \left (2 m^{2}+3\right ) {\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )^{2}}{m^{4}+10 m^{2}+9}+\frac {6 \left (2 m^{2}+3\right ) {\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )^{4}}{m^{4}+10 m^{2}+9}+\frac {8 m \left (m^{2}+4\right ) {\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )^{3}}{m^{4}+10 m^{2}+9}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(195\) |
orering | \(\frac {4 m \left (m^{2}+5\right ) {\mathrm e}^{m x} \sin \left (x \right )^{3}}{m^{4}+10 m^{2}+9}-\frac {2 \left (3 m^{2}+5\right ) \left (m \,{\mathrm e}^{m x} \sin \left (x \right )^{3}+3 \,{\mathrm e}^{m x} \sin \left (x \right )^{2} \cos \left (x \right )\right )}{m^{4}+10 m^{2}+9}+\frac {4 m \left (m^{2} {\mathrm e}^{m x} \sin \left (x \right )^{3}+6 m \,{\mathrm e}^{m x} \sin \left (x \right )^{2} \cos \left (x \right )+6 \,{\mathrm e}^{m x} \sin \left (x \right ) \cos \left (x \right )^{2}-3 \,{\mathrm e}^{m x} \sin \left (x \right )^{3}\right )}{m^{4}+10 m^{2}+9}-\frac {m^{3} {\mathrm e}^{m x} \sin \left (x \right )^{3}+9 m^{2} {\mathrm e}^{m x} \sin \left (x \right )^{2} \cos \left (x \right )+18 m \,{\mathrm e}^{m x} \sin \left (x \right ) \cos \left (x \right )^{2}-9 m \,{\mathrm e}^{m x} \sin \left (x \right )^{3}+6 \,{\mathrm e}^{m x} \cos \left (x \right )^{3}-21 \,{\mathrm e}^{m x} \sin \left (x \right )^{2} \cos \left (x \right )}{m^{4}+10 m^{2}+9}\) | \(225\) |
Input:
int(exp(m*x)*sin(x)^3,x,method=_RETURNVERBOSE)
Output:
1/8*I/(3*I+m)*exp((3*I+m)*x)-3/8*I/(m+I)*exp(x*(m+I))+3/8*I/(-I+m)*exp(x*( -I+m))-1/8*I/(m-3*I)*exp(x*(m-3*I))
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int e^{m x} \sin ^3(x) \, dx=\frac {{\left (m^{3} - {\left (m^{3} + m\right )} \cos \left (x\right )^{2} + 7 \, m\right )} e^{\left (m x\right )} \sin \left (x\right ) + 3 \, {\left ({\left (m^{2} + 1\right )} \cos \left (x\right )^{3} - {\left (m^{2} + 3\right )} \cos \left (x\right )\right )} e^{\left (m x\right )}}{m^{4} + 10 \, m^{2} + 9} \] Input:
integrate(exp(m*x)*sin(x)^3,x, algorithm="fricas")
Output:
((m^3 - (m^3 + m)*cos(x)^2 + 7*m)*e^(m*x)*sin(x) + 3*((m^2 + 1)*cos(x)^3 - (m^2 + 3)*cos(x))*e^(m*x))/(m^4 + 10*m^2 + 9)
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 638, normalized size of antiderivative = 7.78 \[ \int e^{m x} \sin ^3(x) \, dx =\text {Too large to display} \] Input:
integrate(exp(m*x)*sin(x)**3,x)
Output:
Piecewise((x*exp(-3*I*x)*sin(x)**3/8 - 3*I*x*exp(-3*I*x)*sin(x)**2*cos(x)/ 8 - 3*x*exp(-3*I*x)*sin(x)*cos(x)**2/8 + I*x*exp(-3*I*x)*cos(x)**3/8 + 7*I *exp(-3*I*x)*sin(x)**3/24 + I*exp(-3*I*x)*sin(x)*cos(x)**2/4 + exp(-3*I*x) *cos(x)**3/8, Eq(m, -3*I)), (3*x*exp(-I*x)*sin(x)**3/8 - 3*I*x*exp(-I*x)*s in(x)**2*cos(x)/8 + 3*x*exp(-I*x)*sin(x)*cos(x)**2/8 - 3*I*x*exp(-I*x)*cos (x)**3/8 + 5*I*exp(-I*x)*sin(x)**3/8 + 3*I*exp(-I*x)*sin(x)*cos(x)**2/4 + 3*exp(-I*x)*cos(x)**3/8, Eq(m, -I)), (3*x*exp(I*x)*sin(x)**3/8 + 3*I*x*exp (I*x)*sin(x)**2*cos(x)/8 + 3*x*exp(I*x)*sin(x)*cos(x)**2/8 + 3*I*x*exp(I*x )*cos(x)**3/8 - 5*I*exp(I*x)*sin(x)**3/8 - 3*I*exp(I*x)*sin(x)*cos(x)**2/4 + 3*exp(I*x)*cos(x)**3/8, Eq(m, I)), (x*exp(3*I*x)*sin(x)**3/8 + 3*I*x*ex p(3*I*x)*sin(x)**2*cos(x)/8 - 3*x*exp(3*I*x)*sin(x)*cos(x)**2/8 - I*x*exp( 3*I*x)*cos(x)**3/8 - 7*I*exp(3*I*x)*sin(x)**3/24 - I*exp(3*I*x)*sin(x)*cos (x)**2/4 + exp(3*I*x)*cos(x)**3/8, Eq(m, 3*I)), (m**3*exp(m*x)*sin(x)**3/( m**4 + 10*m**2 + 9) - 3*m**2*exp(m*x)*sin(x)**2*cos(x)/(m**4 + 10*m**2 + 9 ) + 7*m*exp(m*x)*sin(x)**3/(m**4 + 10*m**2 + 9) + 6*m*exp(m*x)*sin(x)*cos( x)**2/(m**4 + 10*m**2 + 9) - 9*exp(m*x)*sin(x)**2*cos(x)/(m**4 + 10*m**2 + 9) - 6*exp(m*x)*cos(x)**3/(m**4 + 10*m**2 + 9), True))
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int e^{m x} \sin ^3(x) \, dx=\frac {3 \, {\left (m^{2} + 1\right )} \cos \left (3 \, x\right ) e^{\left (m x\right )} - 3 \, {\left (m^{2} + 9\right )} \cos \left (x\right ) e^{\left (m x\right )} - {\left (m^{3} + m\right )} e^{\left (m x\right )} \sin \left (3 \, x\right ) + 3 \, {\left (m^{3} + 9 \, m\right )} e^{\left (m x\right )} \sin \left (x\right )}{4 \, {\left (m^{4} + 10 \, m^{2} + 9\right )}} \] Input:
integrate(exp(m*x)*sin(x)^3,x, algorithm="maxima")
Output:
1/4*(3*(m^2 + 1)*cos(3*x)*e^(m*x) - 3*(m^2 + 9)*cos(x)*e^(m*x) - (m^3 + m) *e^(m*x)*sin(3*x) + 3*(m^3 + 9*m)*e^(m*x)*sin(x))/(m^4 + 10*m^2 + 9)
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int e^{m x} \sin ^3(x) \, dx=-\frac {1}{4} \, {\left (\frac {m \sin \left (3 \, x\right )}{m^{2} + 9} - \frac {3 \, \cos \left (3 \, x\right )}{m^{2} + 9}\right )} e^{\left (m x\right )} + \frac {3}{4} \, {\left (\frac {m \sin \left (x\right )}{m^{2} + 1} - \frac {\cos \left (x\right )}{m^{2} + 1}\right )} e^{\left (m x\right )} \] Input:
integrate(exp(m*x)*sin(x)^3,x, algorithm="giac")
Output:
-1/4*(m*sin(3*x)/(m^2 + 9) - 3*cos(3*x)/(m^2 + 9))*e^(m*x) + 3/4*(m*sin(x) /(m^2 + 1) - cos(x)/(m^2 + 1))*e^(m*x)
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.57 \[ \int e^{m x} \sin ^3(x) \, dx=-\frac {{\mathrm {e}}^{m\,x}\,\left (\frac {3\,\left (\cos \left (x\right )-m\,\sin \left (x\right )\right )}{m^2+1}-\frac {3\,\cos \left (3\,x\right )-m\,\sin \left (3\,x\right )}{m^2+9}\right )}{4} \] Input:
int(exp(m*x)*sin(x)^3,x)
Output:
-(exp(m*x)*((3*(cos(x) - m*sin(x)))/(m^2 + 1) - (3*cos(3*x) - m*sin(3*x))/ (m^2 + 9)))/4
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74 \[ \int e^{m x} \sin ^3(x) \, dx=\frac {e^{m x} \left (-3 \cos \left (x \right ) \sin \left (x \right )^{2} m^{2}-3 \cos \left (x \right ) \sin \left (x \right )^{2}-6 \cos \left (x \right )+\sin \left (x \right )^{3} m^{3}+\sin \left (x \right )^{3} m +6 \sin \left (x \right ) m \right )}{m^{4}+10 m^{2}+9} \] Input:
int(exp(m*x)*sin(x)^3,x)
Output:
(e**(m*x)*( - 3*cos(x)*sin(x)**2*m**2 - 3*cos(x)*sin(x)**2 - 6*cos(x) + si n(x)**3*m**3 + sin(x)**3*m + 6*sin(x)*m))/(m**4 + 10*m**2 + 9)