3.46 \(\int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx\)

Optimal. Leaf size=129 \[ \frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )} \]

[Out]

-3/16*d*exp(b*x+a)*cos(2*d*x+2*c)/(b^2+4*d^2)+3/16*d*exp(b*x+a)*cos(6*d*x+6*c)/(b^2+36*d^2)+3/32*b*exp(b*x+a)*
sin(2*d*x+2*c)/(b^2+4*d^2)-1/32*b*exp(b*x+a)*sin(6*d*x+6*c)/(b^2+36*d^2)

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Rubi [A]  time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4469, 4432} \[ \frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Cos[c + d*x]^3*Sin[c + d*x]^3,x]

[Out]

(-3*d*E^(a + b*x)*Cos[2*c + 2*d*x])/(16*(b^2 + 4*d^2)) + (3*d*E^(a + b*x)*Cos[6*c + 6*d*x])/(16*(b^2 + 36*d^2)
) + (3*b*E^(a + b*x)*Sin[2*c + 2*d*x])/(32*(b^2 + 4*d^2)) - (b*E^(a + b*x)*Sin[6*c + 6*d*x])/(32*(b^2 + 36*d^2
))

Rule 4432

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

Rule 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx &=\int \left (\frac {3}{32} e^{a+b x} \sin (2 c+2 d x)-\frac {1}{32} e^{a+b x} \sin (6 c+6 d x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int e^{a+b x} \sin (6 c+6 d x) \, dx\right )+\frac {3}{32} \int e^{a+b x} \sin (2 c+2 d x) \, dx\\ &=-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )}+\frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.94, size = 111, normalized size = 0.86 \[ \frac {e^{a+b x} \left (-6 d \left (b^2+36 d^2\right ) \cos (2 (c+d x))+6 d \left (b^2+4 d^2\right ) \cos (6 (c+d x))-2 b \sin (2 (c+d x)) \left (\left (b^2+4 d^2\right ) \cos (4 (c+d x))-b^2-52 d^2\right )\right )}{32 \left (b^4+40 b^2 d^2+144 d^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x)*Cos[c + d*x]^3*Sin[c + d*x]^3,x]

[Out]

(E^(a + b*x)*(-6*d*(b^2 + 36*d^2)*Cos[2*(c + d*x)] + 6*d*(b^2 + 4*d^2)*Cos[6*(c + d*x)] - 2*b*(-b^2 - 52*d^2 +
(b^2 + 4*d^2)*Cos[4*(c + d*x)])*Sin[2*(c + d*x)]))/(32*(b^4 + 40*b^2*d^2 + 144*d^4))

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fricas [A]  time = 1.46, size = 156, normalized size = 1.21 \[ -\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{5} - 6 \, b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - 3 \, {\left (2 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{6} + b^{2} d \cos \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + 2 \, d^{3}\right )} e^{\left (b x + a\right )}}{b^{4} + 40 \, b^{2} d^{2} + 144 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cos(d*x+c)^3*sin(d*x+c)^3,x, algorithm="fricas")

[Out]

-(((b^3 + 4*b*d^2)*cos(d*x + c)^5 - 6*b*d^2*cos(d*x + c) - (b^3 + 4*b*d^2)*cos(d*x + c)^3)*e^(b*x + a)*sin(d*x
+ c) - 3*(2*(b^2*d + 4*d^3)*cos(d*x + c)^6 + b^2*d*cos(d*x + c)^2 - 3*(b^2*d + 4*d^3)*cos(d*x + c)^4 + 2*d^3)
*e^(b*x + a))/(b^4 + 40*b^2*d^2 + 144*d^4)

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giac [A]  time = 0.16, size = 111, normalized size = 0.86 \[ \frac {1}{32} \, {\left (\frac {6 \, d \cos \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {3}{32} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cos(d*x+c)^3*sin(d*x+c)^3,x, algorithm="giac")

[Out]

1/32*(6*d*cos(6*d*x + 6*c)/(b^2 + 36*d^2) - b*sin(6*d*x + 6*c)/(b^2 + 36*d^2))*e^(b*x + a) - 3/32*(2*d*cos(2*d
*x + 2*c)/(b^2 + 4*d^2) - b*sin(2*d*x + 2*c)/(b^2 + 4*d^2))*e^(b*x + a)

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maple [A]  time = 0.10, size = 118, normalized size = 0.91 \[ -\frac {3 d \,{\mathrm e}^{b x +a} \cos \left (2 d x +2 c \right )}{16 \left (b^{2}+4 d^{2}\right )}+\frac {3 d \,{\mathrm e}^{b x +a} \cos \left (6 d x +6 c \right )}{16 \left (b^{2}+36 d^{2}\right )}+\frac {3 b \,{\mathrm e}^{b x +a} \sin \left (2 d x +2 c \right )}{32 \left (b^{2}+4 d^{2}\right )}-\frac {b \,{\mathrm e}^{b x +a} \sin \left (6 d x +6 c \right )}{32 \left (b^{2}+36 d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*cos(d*x+c)^3*sin(d*x+c)^3,x)

[Out]

-3/16*d*exp(b*x+a)*cos(2*d*x+2*c)/(b^2+4*d^2)+3/16*d*exp(b*x+a)*cos(6*d*x+6*c)/(b^2+36*d^2)+3/32*b*exp(b*x+a)*
sin(2*d*x+2*c)/(b^2+4*d^2)-1/32*b*exp(b*x+a)*sin(6*d*x+6*c)/(b^2+36*d^2)

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maxima [B]  time = 0.36, size = 550, normalized size = 4.26 \[ \frac {{\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x\right ) e^{\left (b x\right )} + {\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x + 12 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x + 8 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x - 4 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x\right ) - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x + 12 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x + 8 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x - 4 \, c\right )}{64 \, {\left (b^{4} \cos \left (6 \, c\right )^{2} + b^{4} \sin \left (6 \, c\right )^{2} + 144 \, {\left (\cos \left (6 \, c\right )^{2} + \sin \left (6 \, c\right )^{2}\right )} d^{4} + 40 \, {\left (b^{2} \cos \left (6 \, c\right )^{2} + b^{2} \sin \left (6 \, c\right )^{2}\right )} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*cos(d*x+c)^3*sin(d*x+c)^3,x, algorithm="maxima")

[Out]

1/64*((6*b^2*d*cos(6*c)*e^a + 24*d^3*cos(6*c)*e^a - b^3*e^a*sin(6*c) - 4*b*d^2*e^a*sin(6*c))*cos(6*d*x)*e^(b*x
) + (6*b^2*d*cos(6*c)*e^a + 24*d^3*cos(6*c)*e^a + b^3*e^a*sin(6*c) + 4*b*d^2*e^a*sin(6*c))*cos(6*d*x + 12*c)*e
^(b*x) - 3*(2*b^2*d*cos(6*c)*e^a + 72*d^3*cos(6*c)*e^a + b^3*e^a*sin(6*c) + 36*b*d^2*e^a*sin(6*c))*cos(2*d*x +
8*c)*e^(b*x) - 3*(2*b^2*d*cos(6*c)*e^a + 72*d^3*cos(6*c)*e^a - b^3*e^a*sin(6*c) - 36*b*d^2*e^a*sin(6*c))*cos(
2*d*x - 4*c)*e^(b*x) - (b^3*cos(6*c)*e^a + 4*b*d^2*cos(6*c)*e^a + 6*b^2*d*e^a*sin(6*c) + 24*d^3*e^a*sin(6*c))*
e^(b*x)*sin(6*d*x) - (b^3*cos(6*c)*e^a + 4*b*d^2*cos(6*c)*e^a - 6*b^2*d*e^a*sin(6*c) - 24*d^3*e^a*sin(6*c))*e^
(b*x)*sin(6*d*x + 12*c) + 3*(b^3*cos(6*c)*e^a + 36*b*d^2*cos(6*c)*e^a - 2*b^2*d*e^a*sin(6*c) - 72*d^3*e^a*sin(
6*c))*e^(b*x)*sin(2*d*x + 8*c) + 3*(b^3*cos(6*c)*e^a + 36*b*d^2*cos(6*c)*e^a + 2*b^2*d*e^a*sin(6*c) + 72*d^3*e
^a*sin(6*c))*e^(b*x)*sin(2*d*x - 4*c))/(b^4*cos(6*c)^2 + b^4*sin(6*c)^2 + 144*(cos(6*c)^2 + sin(6*c)^2)*d^4 +
40*(b^2*cos(6*c)^2 + b^2*sin(6*c)^2)*d^2)

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mupad [B]  time = 1.01, size = 178, normalized size = 1.38 \[ -\frac {3\,{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )-\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )-\sin \left (6\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (6\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )+\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )+\sin \left (6\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,\left (b+d\,6{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*exp(a + b*x)*sin(c + d*x)^3,x)

[Out]

(exp(a + b*x)*(cos(6*d*x) - sin(6*d*x)*1i)*(cos(6*c) - sin(6*c)*1i))/(64*(b*1i + 6*d)) - (3*exp(a + b*x)*(cos(
2*d*x) - sin(2*d*x)*1i)*(cos(2*c) - sin(2*c)*1i))/(64*(b*1i + 2*d)) - (exp(a + b*x)*(cos(2*d*x) + sin(2*d*x)*1
i)*(cos(2*c) + sin(2*c)*1i)*3i)/(64*(b + d*2i)) + (exp(a + b*x)*(cos(6*d*x) + sin(6*d*x)*1i)*(cos(6*c) + sin(6
*c)*1i)*1i)/(64*(b + d*6i))

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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(exp(b*x+a)*cos(d*x+c)**3*sin(d*x+c)**3,x)

[Out]

Timed out

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