3.275 \(\int e^{\frac {x}{2}+x z} x^4 \sin ^4(\pi z) \, dz\)

Optimal. Leaf size=199 \[ \frac {x^5 e^{x z+\frac {x}{2}} \sin ^4(\pi z)}{x^2+16 \pi ^2}-\frac {4 \pi x^4 e^{x z+\frac {x}{2}} \sin ^3(\pi z) \cos (\pi z)}{x^2+16 \pi ^2}-\frac {24 \pi ^3 x^4 e^{x z+\frac {x}{2}} \sin (\pi z) \cos (\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4}+\frac {12 \pi ^2 x^5 e^{x z+\frac {x}{2}} \sin ^2(\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4}+\frac {24 \pi ^4 x^3 e^{x z+\frac {x}{2}}}{x^4+20 \pi ^2 x^2+64 \pi ^4} \]

[Out]

24*exp(1/2*x+x*z)*Pi^4*x^3/(64*Pi^4+20*Pi^2*x^2+x^4)-24*exp(1/2*x+x*z)*Pi^3*x^4*cos(Pi*z)*sin(Pi*z)/(64*Pi^4+2
0*Pi^2*x^2+x^4)+12*exp(1/2*x+x*z)*Pi^2*x^5*sin(Pi*z)^2/(64*Pi^4+20*Pi^2*x^2+x^4)-4*exp(1/2*x+x*z)*Pi*x^4*cos(P
i*z)*sin(Pi*z)^3/(16*Pi^2+x^2)+exp(1/2*x+x*z)*x^5*sin(Pi*z)^4/(16*Pi^2+x^2)

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Rubi [A]  time = 0.10, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 4434, 2194} \[ \frac {24 \pi ^4 x^3 e^{x z+\frac {x}{2}}}{x^4+20 \pi ^2 x^2+64 \pi ^4}+\frac {x^5 e^{x z+\frac {x}{2}} \sin ^4(\pi z)}{x^2+16 \pi ^2}+\frac {12 \pi ^2 x^5 e^{x z+\frac {x}{2}} \sin ^2(\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4}-\frac {4 \pi x^4 e^{x z+\frac {x}{2}} \sin ^3(\pi z) \cos (\pi z)}{x^2+16 \pi ^2}-\frac {24 \pi ^3 x^4 e^{x z+\frac {x}{2}} \sin (\pi z) \cos (\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(x/2 + x*z)*x^4*Sin[Pi*z]^4,z]

[Out]

(24*E^(x/2 + x*z)*Pi^4*x^3)/(64*Pi^4 + 20*Pi^2*x^2 + x^4) - (24*E^(x/2 + x*z)*Pi^3*x^4*Cos[Pi*z]*Sin[Pi*z])/(6
4*Pi^4 + 20*Pi^2*x^2 + x^4) + (12*E^(x/2 + x*z)*Pi^2*x^5*Sin[Pi*z]^2)/(64*Pi^4 + 20*Pi^2*x^2 + x^4) - (4*E^(x/
2 + x*z)*Pi*x^4*Cos[Pi*z]*Sin[Pi*z]^3)/(16*Pi^2 + x^2) + (E^(x/2 + x*z)*x^5*Sin[Pi*z]^4)/(16*Pi^2 + x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 4434

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Sin[d + e*x]^n)/(e^2*n^2 + b^2*c^2*Log[F]^2), x] + (Dist[(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] - 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]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
n, 1]

Rubi steps

\begin {align*} \int e^{\frac {x}{2}+x z} x^4 \sin ^4(\pi z) \, dz &=x^4 \int e^{\frac {x}{2}+x z} \sin ^4(\pi z) \, dz\\ &=-\frac {4 e^{\frac {x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac {e^{\frac {x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}+\frac {\left (12 \pi ^2 x^4\right ) \int e^{\frac {x}{2}+x z} \sin ^2(\pi z) \, dz}{16 \pi ^2+x^2}\\ &=-\frac {24 e^{\frac {x}{2}+x z} \pi ^3 x^4 \cos (\pi z) \sin (\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}+\frac {12 e^{\frac {x}{2}+x z} \pi ^2 x^5 \sin ^2(\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac {4 e^{\frac {x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac {e^{\frac {x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}+\frac {\left (24 \pi ^4 x^4\right ) \int e^{\frac {x}{2}+x z} \, dz}{64 \pi ^4+20 \pi ^2 x^2+x^4}\\ &=\frac {24 e^{\frac {x}{2}+x z} \pi ^4 x^3}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac {24 e^{\frac {x}{2}+x z} \pi ^3 x^4 \cos (\pi z) \sin (\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}+\frac {12 e^{\frac {x}{2}+x z} \pi ^2 x^5 \sin ^2(\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac {4 e^{\frac {x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac {e^{\frac {x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 136, normalized size = 0.68 \[ \frac {x^4 e^{x \left (z+\frac {1}{2}\right )} \left (3 x^4-8 \pi x^3 \sin (2 \pi z)+4 \pi x^3 \sin (4 \pi z)-4 \left (x^2+16 \pi ^2\right ) x^2 \cos (2 \pi z)+\left (x^2+4 \pi ^2\right ) x^2 \cos (4 \pi z)+60 \pi ^2 x^2-128 \pi ^3 x \sin (2 \pi z)+16 \pi ^3 x \sin (4 \pi z)+192 \pi ^4\right )}{8 \left (x^5+20 \pi ^2 x^3+64 \pi ^4 x\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(x/2 + x*z)*x^4*Sin[Pi*z]^4,z]

[Out]

(E^(x*(1/2 + z))*x^4*(192*Pi^4 + 60*Pi^2*x^2 + 3*x^4 - 4*x^2*(16*Pi^2 + x^2)*Cos[2*Pi*z] + x^2*(4*Pi^2 + x^2)*
Cos[4*Pi*z] - 128*Pi^3*x*Sin[2*Pi*z] - 8*Pi*x^3*Sin[2*Pi*z] + 16*Pi^3*x*Sin[4*Pi*z] + 4*Pi*x^3*Sin[4*Pi*z]))/(
8*(64*Pi^4*x + 20*Pi^2*x^3 + x^5))

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fricas [A]  time = 0.44, size = 145, normalized size = 0.73 \[ \frac {4 \, {\left ({\left (4 \, \pi ^{3} x^{4} + \pi x^{6}\right )} \cos \left (\pi z\right )^{3} - {\left (10 \, \pi ^{3} x^{4} + \pi x^{6}\right )} \cos \left (\pi z\right )\right )} e^{\left (x z + \frac {1}{2} \, x\right )} \sin \left (\pi z\right ) + {\left (24 \, \pi ^{4} x^{3} + 16 \, \pi ^{2} x^{5} + x^{7} + {\left (4 \, \pi ^{2} x^{5} + x^{7}\right )} \cos \left (\pi z\right )^{4} - 2 \, {\left (10 \, \pi ^{2} x^{5} + x^{7}\right )} \cos \left (\pi z\right )^{2}\right )} e^{\left (x z + \frac {1}{2} \, x\right )}}{64 \, \pi ^{4} + 20 \, \pi ^{2} x^{2} + x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(1/2*x+x*z)*sin(pi*z)^4,z, algorithm="fricas")

[Out]

(4*((4*pi^3*x^4 + pi*x^6)*cos(pi*z)^3 - (10*pi^3*x^4 + pi*x^6)*cos(pi*z))*e^(x*z + 1/2*x)*sin(pi*z) + (24*pi^4
*x^3 + 16*pi^2*x^5 + x^7 + (4*pi^2*x^5 + x^7)*cos(pi*z)^4 - 2*(10*pi^2*x^5 + x^7)*cos(pi*z)^2)*e^(x*z + 1/2*x)
)/(64*pi^4 + 20*pi^2*x^2 + x^4)

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giac [A]  time = 1.01, size = 114, normalized size = 0.57 \[ \frac {1}{8} \, {\left ({\left (\frac {x \cos \left (4 \, \pi z\right )}{16 \, \pi ^{2} + x^{2}} + \frac {4 \, \pi \sin \left (4 \, \pi z\right )}{16 \, \pi ^{2} + x^{2}}\right )} e^{\left (x z + \frac {1}{2} \, x\right )} - 4 \, {\left (\frac {x \cos \left (2 \, \pi z\right )}{4 \, \pi ^{2} + x^{2}} + \frac {2 \, \pi \sin \left (2 \, \pi z\right )}{4 \, \pi ^{2} + x^{2}}\right )} e^{\left (x z + \frac {1}{2} \, x\right )} + \frac {3 \, e^{\left (x z + \frac {1}{2} \, x\right )}}{x}\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(1/2*x+x*z)*sin(pi*z)^4,z, algorithm="giac")

[Out]

1/8*((x*cos(4*pi*z)/(16*pi^2 + x^2) + 4*pi*sin(4*pi*z)/(16*pi^2 + x^2))*e^(x*z + 1/2*x) - 4*(x*cos(2*pi*z)/(4*
pi^2 + x^2) + 2*pi*sin(2*pi*z)/(4*pi^2 + x^2))*e^(x*z + 1/2*x) + 3*e^(x*z + 1/2*x)/x)*x^4

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maple [A]  time = 0.10, size = 127, normalized size = 0.64 \[ \frac {\left (-\frac {4 x \cos \left (2 \pi z \right ) {\mathrm e}^{x z +\frac {1}{2} x}}{x^{2}+4 \pi ^{2}}+\frac {x \cos \left (4 \pi z \right ) {\mathrm e}^{x z +\frac {1}{2} x}}{x^{2}+16 \pi ^{2}}-\frac {8 \pi \,{\mathrm e}^{x z +\frac {1}{2} x} \sin \left (2 \pi z \right )}{x^{2}+4 \pi ^{2}}+\frac {4 \pi \,{\mathrm e}^{x z +\frac {1}{2} x} \sin \left (4 \pi z \right )}{x^{2}+16 \pi ^{2}}+\frac {3 \,{\mathrm e}^{x z +\frac {1}{2} x}}{x}\right ) x^{4}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*exp(1/2*x+x*z)*sin(Pi*z)^4,z)

[Out]

1/8*x^4*(3/x*exp(1/2*x+x*z)+x/(16*Pi^2+x^2)*exp(1/2*x+x*z)*cos(4*Pi*z)+4*Pi/(16*Pi^2+x^2)*exp(1/2*x+x*z)*sin(4
*Pi*z)-4*x/(4*Pi^2+x^2)*exp(1/2*x+x*z)*cos(2*Pi*z)-8*Pi/(4*Pi^2+x^2)*exp(1/2*x+x*z)*sin(2*Pi*z))

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maxima [A]  time = 0.54, size = 160, normalized size = 0.80 \[ \frac {{\left ({\left (4 \, \pi ^{2} x^{2} + x^{4}\right )} \cos \left (4 \, \pi z\right ) e^{\left (x z + \frac {1}{2} \, x\right )} - 4 \, {\left (16 \, \pi ^{2} x^{2} + x^{4}\right )} \cos \left (2 \, \pi z\right ) e^{\left (x z + \frac {1}{2} \, x\right )} + 4 \, {\left (4 \, \pi ^{3} x + \pi x^{3}\right )} e^{\left (x z + \frac {1}{2} \, x\right )} \sin \left (4 \, \pi z\right ) - 8 \, {\left (16 \, \pi ^{3} x + \pi x^{3}\right )} e^{\left (x z + \frac {1}{2} \, x\right )} \sin \left (2 \, \pi z\right ) + 3 \, {\left (64 \, \pi ^{4} + 20 \, \pi ^{2} x^{2} + x^{4}\right )} e^{\left (x z + \frac {1}{2} \, x\right )}\right )} x^{4}}{8 \, {\left (64 \, \pi ^{4} x + 20 \, \pi ^{2} x^{3} + x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*exp(1/2*x+x*z)*sin(pi*z)^4,z, algorithm="maxima")

[Out]

1/8*((4*pi^2*x^2 + x^4)*cos(4*pi*z)*e^(x*z + 1/2*x) - 4*(16*pi^2*x^2 + x^4)*cos(2*pi*z)*e^(x*z + 1/2*x) + 4*(4
*pi^3*x + pi*x^3)*e^(x*z + 1/2*x)*sin(4*pi*z) - 8*(16*pi^3*x + pi*x^3)*e^(x*z + 1/2*x)*sin(2*pi*z) + 3*(64*pi^
4 + 20*pi^2*x^2 + x^4)*e^(x*z + 1/2*x))*x^4/(64*pi^4*x + 20*pi^2*x^3 + x^5)

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mupad [B]  time = 1.11, size = 140, normalized size = 0.70 \[ \frac {x^3\,{\mathrm {e}}^{\frac {x}{2}+x\,z}\,\left (24\,\Pi ^4-\frac {x^4\,\cos \left (2\,\Pi \,z\right )}{2}+\frac {x^4\,\cos \left (4\,\Pi \,z\right )}{8}+\frac {3\,x^4}{8}+\frac {15\,\Pi ^2\,x^2}{2}-\Pi \,x^3\,\sin \left (2\,\Pi \,z\right )-16\,\Pi ^3\,x\,\sin \left (2\,\Pi \,z\right )+\frac {\Pi \,x^3\,\sin \left (4\,\Pi \,z\right )}{2}+2\,\Pi ^3\,x\,\sin \left (4\,\Pi \,z\right )-8\,\Pi ^2\,x^2\,\cos \left (2\,\Pi \,z\right )+\frac {\Pi ^2\,x^2\,\cos \left (4\,\Pi \,z\right )}{2}\right )}{64\,\Pi ^4+20\,\Pi ^2\,x^2+x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*exp(x/2 + x*z)*sin(Pi*z)^4,z)

[Out]

(x^3*exp(x/2 + x*z)*(24*Pi^4 - (x^4*cos(2*Pi*z))/2 + (x^4*cos(4*Pi*z))/8 + (3*x^4)/8 + (15*Pi^2*x^2)/2 - Pi*x^
3*sin(2*Pi*z) - 16*Pi^3*x*sin(2*Pi*z) + (Pi*x^3*sin(4*Pi*z))/2 + 2*Pi^3*x*sin(4*Pi*z) - 8*Pi^2*x^2*cos(2*Pi*z)
+ (Pi^2*x^2*cos(4*Pi*z))/2))/(64*Pi^4 + x^4 + 20*Pi^2*x^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*exp(1/2*x+x*z)*sin(pi*z)**4,z)

[Out]

Timed out

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