\(\int \cos ^3(c+d x) \sin ^3(a+b x) \, dx\) [226]

Optimal result

Integrand size = 17, antiderivative size = 195 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac {9 \cos (a-c+(b-d) x)}{32 (b-d)}+\frac {\cos (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac {3 \cos (3 a-c+(3 b-d) x)}{32 (3 b-d)}-\frac {9 \cos (a+c+(b+d) x)}{32 (b+d)}+\frac {\cos (3 (a+c)+3 (b+d) x)}{96 (b+d)}+\frac {3 \cos (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac {3 \cos (a+3 c+(b+3 d) x)}{32 (b+3 d)} \]

[Out]

-3/32*cos(a-3*c+(b-3*d)*x)/(b-3*d)-9/32*cos(a-c+(b-d)*x)/(b-d)+1/96*cos(3*a-3*c+3*(b-d)*x)/(b-d)+3/32*cos(3*a-
c+(3*b-d)*x)/(3*b-d)-9/32*cos(a+c+(b+d)*x)/(b+d)+1/96*cos(3*a+3*c+3*(b+d)*x)/(b+d)+3/32*cos(3*a+c+(3*b+d)*x)/(
3*b+d)-3/32*cos(a+3*c+(b+3*d)*x)/(b+3*d)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4670, 2718} \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac {9 \cos (a+x (b-d)-c)}{32 (b-d)}+\frac {\cos (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \cos (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac {9 \cos (a+x (b+d)+c)}{32 (b+d)}+\frac {\cos (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \cos (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac {3 \cos (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]

[In]

Int[Cos[c + d*x]^3*Sin[a + b*x]^3,x]

[Out]

(-3*Cos[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Cos[a - c + (b - d)*x])/(32*(b - d)) + Cos[3*(a - c) + 3*(
b - d)*x]/(96*(b - d)) + (3*Cos[3*a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cos[a + c + (b + d)*x])/(32*(b + d
)) + Cos[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Cos[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Cos[a + 3*
c + (b + 3*d)*x])/(32*(b + 3*d))

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4670

Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
x]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} \sin (a-3 c+(b-3 d) x)+\frac {9}{32} \sin (a-c+(b-d) x)-\frac {1}{32} \sin (3 (a-c)+3 (b-d) x)-\frac {3}{32} \sin (3 a-c+(3 b-d) x)+\frac {9}{32} \sin (a+c+(b+d) x)-\frac {1}{32} \sin (3 (a+c)+3 (b+d) x)-\frac {3}{32} \sin (3 a+c+(3 b+d) x)+\frac {3}{32} \sin (a+3 c+(b+3 d) x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int \sin (3 (a-c)+3 (b-d) x) \, dx\right )-\frac {1}{32} \int \sin (3 (a+c)+3 (b+d) x) \, dx+\frac {3}{32} \int \sin (a-3 c+(b-3 d) x) \, dx-\frac {3}{32} \int \sin (3 a-c+(3 b-d) x) \, dx-\frac {3}{32} \int \sin (3 a+c+(3 b+d) x) \, dx+\frac {3}{32} \int \sin (a+3 c+(b+3 d) x) \, dx+\frac {9}{32} \int \sin (a-c+(b-d) x) \, dx+\frac {9}{32} \int \sin (a+c+(b+d) x) \, dx \\ & = -\frac {3 \cos (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac {9 \cos (a-c+(b-d) x)}{32 (b-d)}+\frac {\cos (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac {3 \cos (3 a-c+(3 b-d) x)}{32 (3 b-d)}-\frac {9 \cos (a+c+(b+d) x)}{32 (b+d)}+\frac {\cos (3 (a+c)+3 (b+d) x)}{96 (b+d)}+\frac {3 \cos (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac {3 \cos (a+3 c+(b+3 d) x)}{32 (b+3 d)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {1}{96} \left (-\frac {9 \cos (a-3 c+b x-3 d x)}{b-3 d}-\frac {27 \cos (a-c+b x-d x)}{b-d}+\frac {\cos (3 (a-c+b x-d x))}{b-d}+\frac {9 \cos (3 a-c+3 b x-d x)}{3 b-d}+\frac {9 \cos (3 a+c+3 b x+d x)}{3 b+d}-\frac {9 \cos (a+3 c+b x+3 d x)}{b+3 d}-\frac {27 \cos (a+c+(b+d) x)}{b+d}+\frac {\cos (3 (a+c+(b+d) x))}{b+d}\right ) \]

[In]

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

[Out]

((-9*Cos[a - 3*c + b*x - 3*d*x])/(b - 3*d) - (27*Cos[a - c + b*x - d*x])/(b - d) + Cos[3*(a - c + b*x - d*x)]/
(b - d) + (9*Cos[3*a - c + 3*b*x - d*x])/(3*b - d) + (9*Cos[3*a + c + 3*b*x + d*x])/(3*b + d) - (9*Cos[a + 3*c
+ b*x + 3*d*x])/(b + 3*d) - (27*Cos[a + c + (b + d)*x])/(b + d) + Cos[3*(a + c + (b + d)*x)]/(b + d))/96

Maple [A] (verified)

Time = 2.97 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {3 \cos \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}-\frac {9 \cos \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}-\frac {9 \cos \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}-\frac {3 \cos \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\cos \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}+\frac {3 \cos \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \cos \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\cos \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d}\)	\(190\)
parallelrisch	\(\frac {-36 \left (\left (b^{2}-\frac {61 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d^{2}}{3}+3 b^{2}-7 d^{2}\right ) b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}-216 \left (\left (b^{2}-\frac {7 d^{2}}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{2}-\frac {38 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2}-\frac {7 d^{2}}{3}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-108 b \left (\left (b^{4}-\frac {70}{9} b^{2} d^{2}+\frac {7}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {7}{3} b^{2} d^{2}+d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (3 b^{4}-18 b^{2} d^{2}-d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2} d^{2}-\frac {7 d^{4}}{3}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-576 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (b^{4}-\frac {49}{12} b^{2} d^{2}+\frac {3}{4} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{4}+\frac {23}{18} b^{2} d^{2}+\frac {1}{2} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{4}-\frac {49 b^{2} d^{2}}{12}+\frac {3 d^{4}}{4}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-324 \left (\left (\frac {1}{3} b^{2} d^{2}-\frac {7}{9} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (b^{4}-6 b^{2} d^{2}-\frac {1}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {7}{9} b^{2} d^{2}+\frac {1}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {b^{4}}{3}-\frac {70 b^{2} d^{2}}{27}+\frac {7 d^{4}}{9}\right ) b \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-216 \left (\left (b^{2}-\frac {7 d^{2}}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{2}-\frac {38 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2}-\frac {7 d^{2}}{3}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+\left (-108 b^{5}+252 b^{3} d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+480 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3} d^{2}-36 b^{5}+244 b^{3} d^{2}}{27 \left (b +d \right ) \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3} \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b -\frac {d}{3}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (b -d \right ) \left (b +3 d \right )}\)	\(688\)
risch	\(\text {Expression too large to display}\)	\(1466\)

[In]

int(cos(d*x+c)^3*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-3/32*cos(a-3*c+(b-3*d)*x)/(b-3*d)-9/32*cos(a-c+(b-d)*x)/(b-d)-9/32*cos(a+c+(b+d)*x)/(b+d)-3/32*cos(a+3*c+(b+3
*d)*x)/(b+3*d)+1/32/(3*b-3*d)*cos((3*b-3*d)*x+3*a-3*c)+3/32*cos(3*a-c+(3*b-d)*x)/(3*b-d)+3/32*cos(3*a+c+(3*b+d
)*x)/(3*b+d)+1/32/(3*b+3*d)*cos((3*b+3*d)*x+3*a+3*c)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.35 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {{\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{5} - 28 \, b^{3} d^{2} + 3 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )^{3} + {\left (122 \, b^{2} d^{3} - 18 \, d^{5} - 2 \, {\left (b^{2} d^{3} - 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2} - {\left (63 \, b^{4} d - 88 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 6 \, {\left ({\left (b^{3} d^{2} - 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (7 \, b^{3} d^{2} - 3 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )}{3 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )}} \]

[In]

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

[Out]

1/3*(((9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cos(b*x + a)^3 - 3*(9*b^5 - 28*b^3*d^2 + 3*b*d^4)*cos(b*x + a))*cos(d*x +
c)^3 + (122*b^2*d^3 - 18*d^5 - 2*(b^2*d^3 - 9*d^5)*cos(b*x + a)^2 - (63*b^4*d - 88*b^2*d^3 + 9*d^5 - (9*b^4*d
- 82*b^2*d^3 + 9*d^5)*cos(b*x + a)^2)*cos(d*x + c)^2)*sin(b*x + a)*sin(d*x + c) - 6*((b^3*d^2 - 9*b*d^4)*cos(
b*x + a)^3 - 3*(7*b^3*d^2 - 3*b*d^4)*cos(b*x + a))*cos(d*x + c))/(9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3577 vs. \(2 (172) = 344\).

Time = 17.96 (sec) , antiderivative size = 3577, normalized size of antiderivative = 18.34 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((x*sin(a)**3*cos(c)**3, Eq(b, 0) & Eq(d, 0)), (-9*x*sin(a - 3*d*x)**3*sin(c + d*x)**2*cos(c + d*x)/3
2 + 3*x*sin(a - 3*d*x)**3*cos(c + d*x)**3/32 - 3*x*sin(a - 3*d*x)**2*sin(c + d*x)**3*cos(a - 3*d*x)/32 + 9*x*s
in(a - 3*d*x)**2*sin(c + d*x)*cos(a - 3*d*x)*cos(c + d*x)**2/32 - 9*x*sin(a - 3*d*x)*sin(c + d*x)**2*cos(a - 3
*d*x)**2*cos(c + d*x)/32 + 3*x*sin(a - 3*d*x)*cos(a - 3*d*x)**2*cos(c + d*x)**3/32 - 3*x*sin(c + d*x)**3*cos(a
- 3*d*x)**3/32 + 9*x*sin(c + d*x)*cos(a - 3*d*x)**3*cos(c + d*x)**2/32 - sin(a - 3*d*x)**3*sin(c + d*x)**3/(1
2*d) - 13*sin(a - 3*d*x)**3*sin(c + d*x)*cos(c + d*x)**2/(320*d) + 3*sin(a - 3*d*x)**2*sin(c + d*x)**2*cos(a -
3*d*x)*cos(c + d*x)/(20*d) + 101*sin(a - 3*d*x)**2*cos(a - 3*d*x)*cos(c + d*x)**3/(320*d) - 27*sin(a - 3*d*x)
*sin(c + d*x)**3*cos(a - 3*d*x)**2/(320*d) + 51*sin(c + d*x)**2*cos(a - 3*d*x)**3*cos(c + d*x)/(320*d) + cos(a
- 3*d*x)**3*cos(c + d*x)**3/(5*d), Eq(b, -3*d)), (3*x*sin(a - d*x)**3*sin(c + d*x)**2*cos(c + d*x)/16 + 5*x*s
in(a - d*x)**3*cos(c + d*x)**3/16 + 3*x*sin(a - d*x)**2*sin(c + d*x)**3*cos(a - d*x)/16 + 9*x*sin(a - d*x)**2*
sin(c + d*x)*cos(a - d*x)*cos(c + d*x)**2/16 + 9*x*sin(a - d*x)*sin(c + d*x)**2*cos(a - d*x)**2*cos(c + d*x)/1
6 + 3*x*sin(a - d*x)*cos(a - d*x)**2*cos(c + d*x)**3/16 + 5*x*sin(c + d*x)**3*cos(a - d*x)**3/16 + 3*x*sin(c +
d*x)*cos(a - d*x)**3*cos(c + d*x)**2/16 + sin(a - d*x)**3*sin(c + d*x)**3/(48*d) + sin(a - d*x)**3*sin(c + d*
x)*cos(c + d*x)**2/(2*d) + 3*sin(a - d*x)**2*sin(c + d*x)**2*cos(a - d*x)*cos(c + d*x)/(4*d) + 3*sin(a - d*x)*
*2*cos(a - d*x)*cos(c + d*x)**3/(16*d) + 5*sin(a - d*x)*sin(c + d*x)**3*cos(a - d*x)**2/(16*d) + cos(a - d*x)*
*3*cos(c + d*x)**3/(16*d), Eq(b, -d)), (3*x*sin(a - d*x/3)**3*sin(c + d*x)**2*cos(c + d*x)/32 + 3*x*sin(a - d*
x/3)**3*cos(c + d*x)**3/32 + 9*x*sin(a - d*x/3)**2*sin(c + d*x)**3*cos(a - d*x/3)/32 + 9*x*sin(a - d*x/3)**2*s
in(c + d*x)*cos(a - d*x/3)*cos(c + d*x)**2/32 - 9*x*sin(a - d*x/3)*sin(c + d*x)**2*cos(a - d*x/3)**2*cos(c + d
*x)/32 - 9*x*sin(a - d*x/3)*cos(a - d*x/3)**2*cos(c + d*x)**3/32 - 3*x*sin(c + d*x)**3*cos(a - d*x/3)**3/32 -
3*x*sin(c + d*x)*cos(a - d*x/3)**3*cos(c + d*x)**2/32 + 3*sin(a - d*x/3)**3*sin(c + d*x)**3/(4*d) + 351*sin(a
- d*x/3)**3*sin(c + d*x)*cos(c + d*x)**2/(320*d) - 9*sin(a - d*x/3)**2*sin(c + d*x)**2*cos(a - d*x/3)*cos(c +
d*x)/(20*d) - 183*sin(a - d*x/3)**2*cos(a - d*x/3)*cos(c + d*x)**3/(320*d) + 9*sin(a - d*x/3)*sin(c + d*x)**3*
cos(a - d*x/3)**2/(320*d) - 33*sin(c + d*x)**2*cos(a - d*x/3)**3*cos(c + d*x)/(320*d) - cos(a - d*x/3)**3*cos(
c + d*x)**3/(10*d), Eq(b, -d/3)), (3*x*sin(a + d*x/3)**3*sin(c + d*x)**2*cos(c + d*x)/32 + 3*x*sin(a + d*x/3)*
*3*cos(c + d*x)**3/32 - 9*x*sin(a + d*x/3)**2*sin(c + d*x)**3*cos(a + d*x/3)/32 - 9*x*sin(a + d*x/3)**2*sin(c
+ d*x)*cos(a + d*x/3)*cos(c + d*x)**2/32 - 9*x*sin(a + d*x/3)*sin(c + d*x)**2*cos(a + d*x/3)**2*cos(c + d*x)/3
2 - 9*x*sin(a + d*x/3)*cos(a + d*x/3)**2*cos(c + d*x)**3/32 + 3*x*sin(c + d*x)**3*cos(a + d*x/3)**3/32 + 3*x*s
in(c + d*x)*cos(a + d*x/3)**3*cos(c + d*x)**2/32 + 3*sin(a + d*x/3)**3*sin(c + d*x)**3/(4*d) + 351*sin(a + d*x
/3)**3*sin(c + d*x)*cos(c + d*x)**2/(320*d) + 9*sin(a + d*x/3)**2*sin(c + d*x)**2*cos(a + d*x/3)*cos(c + d*x)/
(20*d) + 183*sin(a + d*x/3)**2*cos(a + d*x/3)*cos(c + d*x)**3/(320*d) + 9*sin(a + d*x/3)*sin(c + d*x)**3*cos(a
+ d*x/3)**2/(320*d) + 33*sin(c + d*x)**2*cos(a + d*x/3)**3*cos(c + d*x)/(320*d) + cos(a + d*x/3)**3*cos(c + d
*x)**3/(10*d), Eq(b, d/3)), (3*x*sin(a + d*x)**3*sin(c + d*x)**2*cos(c + d*x)/16 + 5*x*sin(a + d*x)**3*cos(c +
d*x)**3/16 - 3*x*sin(a + d*x)**2*sin(c + d*x)**3*cos(a + d*x)/16 - 9*x*sin(a + d*x)**2*sin(c + d*x)*cos(a + d
*x)*cos(c + d*x)**2/16 + 9*x*sin(a + d*x)*sin(c + d*x)**2*cos(a + d*x)**2*cos(c + d*x)/16 + 3*x*sin(a + d*x)*c
os(a + d*x)**2*cos(c + d*x)**3/16 - 5*x*sin(c + d*x)**3*cos(a + d*x)**3/16 - 3*x*sin(c + d*x)*cos(a + d*x)**3*
cos(c + d*x)**2/16 + sin(a + d*x)**3*sin(c + d*x)**3/(48*d) + sin(a + d*x)**3*sin(c + d*x)*cos(c + d*x)**2/(2*
d) - 3*sin(a + d*x)**2*sin(c + d*x)**2*cos(a + d*x)*cos(c + d*x)/(4*d) - 3*sin(a + d*x)**2*cos(a + d*x)*cos(c
+ d*x)**3/(16*d) + 5*sin(a + d*x)*sin(c + d*x)**3*cos(a + d*x)**2/(16*d) - cos(a + d*x)**3*cos(c + d*x)**3/(16
*d), Eq(b, d)), (-9*x*sin(a + 3*d*x)**3*sin(c + d*x)**2*cos(c + d*x)/32 + 3*x*sin(a + 3*d*x)**3*cos(c + d*x)**
3/32 + 3*x*sin(a + 3*d*x)**2*sin(c + d*x)**3*cos(a + 3*d*x)/32 - 9*x*sin(a + 3*d*x)**2*sin(c + d*x)*cos(a + 3*
d*x)*cos(c + d*x)**2/32 - 9*x*sin(a + 3*d*x)*sin(c + d*x)**2*cos(a + 3*d*x)**2*cos(c + d*x)/32 + 3*x*sin(a + 3
*d*x)*cos(a + 3*d*x)**2*cos(c + d*x)**3/32 + 3*x*sin(c + d*x)**3*cos(a + 3*d*x)**3/32 - 9*x*sin(c + d*x)*cos(a
+ 3*d*x)**3*cos(c + d*x)**2/32 - sin(a + 3*d*x)**3*sin(c + d*x)**3/(30*d) - 61*sin(a + 3*d*x)**3*sin(c + d*x)
*cos(c + d*x)**2/(320*d) - 117*sin(a + 3*d*x)**2*cos(a + 3*d*x)*cos(c + d*x)**3/(320*d) - 11*sin(a + 3*d*x)*si
n(c + d*x)**3*cos(a + 3*d*x)**2/(320*d) - 3*sin(a + 3*d*x)*sin(c + d*x)*cos(a + 3*d*x)**2*cos(c + d*x)**2/(20*
d) - 3*sin(c + d*x)**2*cos(a + 3*d*x)**3*cos(c + d*x)/(320*d) - cos(a + 3*d*x)**3*cos(c + d*x)**3/(4*d), Eq(b,
3*d)), (-27*b**5*sin(a + b*x)**2*cos(a + b*x)*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d
**6) - 18*b**5*cos(a + b*x)**3*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6) - 63*b**4*d
*sin(a + b*x)**3*sin(c + d*x)*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6) - 54*b**4*d*
sin(a + b*x)*sin(c + d*x)*cos(a + b*x)**2*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6)
+ 126*b**3*d**2*sin(a + b*x)**2*sin(c + d*x)**2*cos(a + b*x)*cos(c + d*x)/(27*b**6 - 273*b**4*d**2 + 273*b**2*
d**4 - 27*d**6) + 210*b**3*d**2*sin(a + b*x)**2*cos(a + b*x)*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b*
*2*d**4 - 27*d**6) + 120*b**3*d**2*sin(c + d*x)**2*cos(a + b*x)**3*cos(c + d*x)/(27*b**6 - 273*b**4*d**2 + 273
*b**2*d**4 - 27*d**6) + 122*b**3*d**2*cos(a + b*x)**3*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4
- 27*d**6) + 122*b**2*d**3*sin(a + b*x)**3*sin(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6
) + 210*b**2*d**3*sin(a + b*x)**3*sin(c + d*x)*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d
**6) + 120*b**2*d**3*sin(a + b*x)*sin(c + d*x)**3*cos(a + b*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 2
7*d**6) + 126*b**2*d**3*sin(a + b*x)*sin(c + d*x)*cos(a + b*x)**2*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 2
73*b**2*d**4 - 27*d**6) - 54*b*d**4*sin(a + b*x)**2*sin(c + d*x)**2*cos(a + b*x)*cos(c + d*x)/(27*b**6 - 273*b
**4*d**2 + 273*b**2*d**4 - 27*d**6) - 63*b*d**4*sin(a + b*x)**2*cos(a + b*x)*cos(c + d*x)**3/(27*b**6 - 273*b*
*4*d**2 + 273*b**2*d**4 - 27*d**6) - 18*d**5*sin(a + b*x)**3*sin(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b*
*2*d**4 - 27*d**6) - 27*d**5*sin(a + b*x)**3*sin(c + d*x)*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*
d**4 - 27*d**6), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2612 vs. \(2 (179) = 358\).

Time = 0.40 (sec) , antiderivative size = 2612, normalized size of antiderivative = 13.39 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/192*(9*(3*b^5*cos(3*c) - b^4*d*cos(3*c) - 30*b^3*d^2*cos(3*c) + 10*b^2*d^3*cos(3*c) + 27*b*d^4*cos(3*c) - 9*
d^5*cos(3*c))*cos((3*b + d)*x + 3*a + 4*c) + 9*(3*b^5*cos(3*c) - b^4*d*cos(3*c) - 30*b^3*d^2*cos(3*c) + 10*b^2
*d^3*cos(3*c) + 27*b*d^4*cos(3*c) - 9*d^5*cos(3*c))*cos((3*b + d)*x + 3*a - 2*c) + 9*(3*b^5*cos(3*c) + b^4*d*c
os(3*c) - 30*b^3*d^2*cos(3*c) - 10*b^2*d^3*cos(3*c) + 27*b*d^4*cos(3*c) + 9*d^5*cos(3*c))*cos(-(3*b - d)*x - 3
*a + 4*c) + 9*(3*b^5*cos(3*c) + b^4*d*cos(3*c) - 30*b^3*d^2*cos(3*c) - 10*b^2*d^3*cos(3*c) + 27*b*d^4*cos(3*c)
+ 9*d^5*cos(3*c))*cos(-(3*b - d)*x - 3*a - 2*c) - 9*(9*b^5*cos(3*c) - 27*b^4*d*cos(3*c) - 10*b^3*d^2*cos(3*c)
+ 30*b^2*d^3*cos(3*c) + b*d^4*cos(3*c) - 3*d^5*cos(3*c))*cos((b + 3*d)*x + a + 6*c) - 9*(9*b^5*cos(3*c) - 27*
b^4*d*cos(3*c) - 10*b^3*d^2*cos(3*c) + 30*b^2*d^3*cos(3*c) + b*d^4*cos(3*c) - 3*d^5*cos(3*c))*cos((b + 3*d)*x
+ a) + (9*b^5*cos(3*c) - 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) + 82*b^2*d^3*cos(3*c) + 9*b*d^4*cos(3*c) - 9*d
^5*cos(3*c))*cos(3*(b + d)*x + 3*a + 6*c) + (9*b^5*cos(3*c) - 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) + 82*b^2*
d^3*cos(3*c) + 9*b*d^4*cos(3*c) - 9*d^5*cos(3*c))*cos(3*(b + d)*x + 3*a) - 27*(9*b^5*cos(3*c) - 9*b^4*d*cos(3*
c) - 82*b^3*d^2*cos(3*c) + 82*b^2*d^3*cos(3*c) + 9*b*d^4*cos(3*c) - 9*d^5*cos(3*c))*cos((b + d)*x + a + 4*c) -
27*(9*b^5*cos(3*c) - 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) + 82*b^2*d^3*cos(3*c) + 9*b*d^4*cos(3*c) - 9*d^5*
cos(3*c))*cos((b + d)*x + a - 2*c) - 27*(9*b^5*cos(3*c) + 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) - 82*b^2*d^3*
cos(3*c) + 9*b*d^4*cos(3*c) + 9*d^5*cos(3*c))*cos(-(b - d)*x - a + 4*c) - 27*(9*b^5*cos(3*c) + 9*b^4*d*cos(3*c
) - 82*b^3*d^2*cos(3*c) - 82*b^2*d^3*cos(3*c) + 9*b*d^4*cos(3*c) + 9*d^5*cos(3*c))*cos(-(b - d)*x - a - 2*c) +
(9*b^5*cos(3*c) + 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) - 82*b^2*d^3*cos(3*c) + 9*b*d^4*cos(3*c) + 9*d^5*cos
(3*c))*cos(-3*(b - d)*x - 3*a + 6*c) + (9*b^5*cos(3*c) + 9*b^4*d*cos(3*c) - 82*b^3*d^2*cos(3*c) - 82*b^2*d^3*c
os(3*c) + 9*b*d^4*cos(3*c) + 9*d^5*cos(3*c))*cos(-3*(b - d)*x - 3*a) - 9*(9*b^5*cos(3*c) + 27*b^4*d*cos(3*c) -
10*b^3*d^2*cos(3*c) - 30*b^2*d^3*cos(3*c) + b*d^4*cos(3*c) + 3*d^5*cos(3*c))*cos(-(b - 3*d)*x - a + 6*c) - 9*
(9*b^5*cos(3*c) + 27*b^4*d*cos(3*c) - 10*b^3*d^2*cos(3*c) - 30*b^2*d^3*cos(3*c) + b*d^4*cos(3*c) + 3*d^5*cos(3
*c))*cos(-(b - 3*d)*x - a) + 9*(3*b^5*sin(3*c) - b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) + 10*b^2*d^3*sin(3*c) +
27*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin((3*b + d)*x + 3*a + 4*c) - 9*(3*b^5*sin(3*c) - b^4*d*sin(3*c) - 30*b^3
*d^2*sin(3*c) + 10*b^2*d^3*sin(3*c) + 27*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin((3*b + d)*x + 3*a - 2*c) + 9*(3*
b^5*sin(3*c) + b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) - 10*b^2*d^3*sin(3*c) + 27*b*d^4*sin(3*c) + 9*d^5*sin(3*c)
)*sin(-(3*b - d)*x - 3*a + 4*c) - 9*(3*b^5*sin(3*c) + b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) - 10*b^2*d^3*sin(3*
c) + 27*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*sin(-(3*b - d)*x - 3*a - 2*c) - 9*(9*b^5*sin(3*c) - 27*b^4*d*sin(3*c)
- 10*b^3*d^2*sin(3*c) + 30*b^2*d^3*sin(3*c) + b*d^4*sin(3*c) - 3*d^5*sin(3*c))*sin((b + 3*d)*x + a + 6*c) + 9
*(9*b^5*sin(3*c) - 27*b^4*d*sin(3*c) - 10*b^3*d^2*sin(3*c) + 30*b^2*d^3*sin(3*c) + b*d^4*sin(3*c) - 3*d^5*sin(
3*c))*sin((b + 3*d)*x + a) + (9*b^5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) +
9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin(3*(b + d)*x + 3*a + 6*c) - (9*b^5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*
d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) + 9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin(3*(b + d)*x + 3*a) - 27*(9*b^5*sin
(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) + 9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin(
(b + d)*x + a + 4*c) + 27*(9*b^5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) + 9*b
*d^4*sin(3*c) - 9*d^5*sin(3*c))*sin((b + d)*x + a - 2*c) - 27*(9*b^5*sin(3*c) + 9*b^4*d*sin(3*c) - 82*b^3*d^2*
sin(3*c) - 82*b^2*d^3*sin(3*c) + 9*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*sin(-(b - d)*x - a + 4*c) + 27*(9*b^5*sin(
3*c) + 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) - 82*b^2*d^3*sin(3*c) + 9*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*sin(-
(b - d)*x - a - 2*c) + (9*b^5*sin(3*c) + 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) - 82*b^2*d^3*sin(3*c) + 9*b*d^
4*sin(3*c) + 9*d^5*sin(3*c))*sin(-3*(b - d)*x - 3*a + 6*c) - (9*b^5*sin(3*c) + 9*b^4*d*sin(3*c) - 82*b^3*d^2*s
in(3*c) - 82*b^2*d^3*sin(3*c) + 9*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*sin(-3*(b - d)*x - 3*a) - 9*(9*b^5*sin(3*c)
+ 27*b^4*d*sin(3*c) - 10*b^3*d^2*sin(3*c) - 30*b^2*d^3*sin(3*c) + b*d^4*sin(3*c) + 3*d^5*sin(3*c))*sin(-(b -
3*d)*x - a + 6*c) + 9*(9*b^5*sin(3*c) + 27*b^4*d*sin(3*c) - 10*b^3*d^2*sin(3*c) - 30*b^2*d^3*sin(3*c) + b*d^4*
sin(3*c) + 3*d^5*sin(3*c))*sin(-(b - 3*d)*x - a))/(9*b^6*cos(3*c)^2 + 9*b^6*sin(3*c)^2 - 9*(cos(3*c)^2 + sin(3
*c)^2)*d^6 + 91*(b^2*cos(3*c)^2 + b^2*sin(3*c)^2)*d^4 - 91*(b^4*cos(3*c)^2 + b^4*sin(3*c)^2)*d^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.93 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {\cos \left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}{96 \, {\left (b + d\right )}} + \frac {3 \, \cos \left (3 \, b x + d x + 3 \, a + c\right )}{32 \, {\left (3 \, b + d\right )}} + \frac {3 \, \cos \left (3 \, b x - d x + 3 \, a - c\right )}{32 \, {\left (3 \, b - d\right )}} + \frac {\cos \left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}{96 \, {\left (b - d\right )}} - \frac {3 \, \cos \left (b x + 3 \, d x + a + 3 \, c\right )}{32 \, {\left (b + 3 \, d\right )}} - \frac {9 \, \cos \left (b x + d x + a + c\right )}{32 \, {\left (b + d\right )}} - \frac {9 \, \cos \left (b x - d x + a - c\right )}{32 \, {\left (b - d\right )}} - \frac {3 \, \cos \left (b x - 3 \, d x + a - 3 \, c\right )}{32 \, {\left (b - 3 \, d\right )}} \]

[In]

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

[Out]

1/96*cos(3*b*x + 3*d*x + 3*a + 3*c)/(b + d) + 3/32*cos(3*b*x + d*x + 3*a + c)/(3*b + d) + 3/32*cos(3*b*x - d*x
+ 3*a - c)/(3*b - d) + 1/96*cos(3*b*x - 3*d*x + 3*a - 3*c)/(b - d) - 3/32*cos(b*x + 3*d*x + a + 3*c)/(b + 3*d
) - 9/32*cos(b*x + d*x + a + c)/(b + d) - 9/32*cos(b*x - d*x + a - c)/(b - d) - 3/32*cos(b*x - 3*d*x + a - 3*c
)/(b - 3*d)

Mupad [B] (verification not implemented)

Time = 25.69 (sec) , antiderivative size = 951, normalized size of antiderivative = 4.88 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=-{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {-b^3-b^2\,d+9\,b\,d^2+9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-b^3+b^2\,d+9\,b\,d^2-9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {-b^3+b^2\,d+9\,b\,d^2-9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-b^3-b^2\,d+9\,b\,d^2+9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right ) \]

[In]

int(cos(c + d*x)^3*sin(a + b*x)^3,x)

[Out]

- exp(a*3i - c*1i + b*x*3i - d*x*1i)*((9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3)/(576*b^4 + 64*d^4 - 640*b^2*d^2) + (
exp(- a*6i - b*x*6i)*(9*b*d^2 + 3*b^2*d - 9*b^3 - 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- a*2i - b*x
*2i)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- a*4i - b*x*4i)*(9*b*d^2
+ 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2)) - exp(a*3i + c*1i + b*x*3i + d*x*1i)*((9*b*d^2
+ 3*b^2*d - 9*b^3 - 3*d^3)/(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- a*6i - b*x*6i)*(9*b*d^2 - 3*b^2*d - 9*b^
3 + 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- a*2i - b*x*2i)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(5
76*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- a*4i - b*x*4i)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^
4 - 640*b^2*d^2)) - exp(a*3i - c*3i + b*x*3i - d*x*3i)*((9*b*d^2 - b^2*d - b^3 + 9*d^3)/(192*b^4 + 1728*d^4 -
1920*b^2*d^2) + (exp(- a*6i - b*x*6i)*(9*b*d^2 + b^2*d - b^3 - 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (
exp(- a*2i - b*x*2i)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- a*4i
- b*x*4i)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2)) - exp(a*3i + c*3i + b*x*
3i + d*x*3i)*((9*b*d^2 + b^2*d - b^3 - 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- a*6i - b*x*6i)*(9*b
*d^2 - b^2*d - b^3 + 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- a*2i - b*x*2i)*(9*b*d^2 + 27*b^2*d -
9*b^3 - 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- a*4i - b*x*4i)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27
*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2))