\(\int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx\) [101]

Optimal result

Integrand size = 26, antiderivative size = 121 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=-\frac {3 F^{c (a+b x)} (2 e \cos (2 d+2 e x)-b c \log (F) \sin (2 d+2 e x))}{32 \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {F^{c (a+b x)} (6 e \cos (6 d+6 e x)-b c \log (F) \sin (6 d+6 e x))}{32 \left (36 e^2+b^2 c^2 \log ^2(F)\right )} \] Output:

-3*F^(c*(b*x+a))*(2*e*cos(2*e*x+2*d)-b*c*ln(F)*sin(2*e*x+2*d))/(128*e^2+32 
*b^2*c^2*ln(F)^2)+F^(c*(b*x+a))*(6*e*cos(6*e*x+6*d)-b*c*ln(F)*sin(6*e*x+6* 
d))/(1152*e^2+32*b^2*c^2*ln(F)^2)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.34 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (6 \cos (6 (d+e x)) \left (4 e^3+b^2 c^2 e \log ^2(F)\right )-6 \cos (2 (d+e x)) \left (36 e^3+b^2 c^2 e \log ^2(F)\right )-2 b c \log (F) \left (-52 e^2-b^2 c^2 \log ^2(F)+\cos (4 (d+e x)) \left (4 e^2+b^2 c^2 \log ^2(F)\right )\right ) \sin (2 (d+e x))\right )}{32 \left (144 e^4+40 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \] Input:

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

Output:

(F^(c*(a + b*x))*(6*Cos[6*(d + e*x)]*(4*e^3 + b^2*c^2*e*Log[F]^2) - 6*Cos[ 
2*(d + e*x)]*(36*e^3 + b^2*c^2*e*Log[F]^2) - 2*b*c*Log[F]*(-52*e^2 - b^2*c 
^2*Log[F]^2 + Cos[4*(d + e*x)]*(4*e^2 + b^2*c^2*Log[F]^2))*Sin[2*(d + e*x) 
]))/(32*(144*e^4 + 40*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4972, 2009}

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.

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

\(\Big \downarrow \) 4972

\(\displaystyle \int \left (\frac {3}{32} \sin (2 d+2 e x) F^{c (a+b x)}-\frac {1}{32} \sin (6 d+6 e x) F^{c (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b c \log (F) \sin (2 d+2 e x) F^{c (a+b x)}}{32 \left (b^2 c^2 \log ^2(F)+4 e^2\right )}-\frac {b c \log (F) \sin (6 d+6 e x) F^{c (a+b x)}}{32 \left (b^2 c^2 \log ^2(F)+36 e^2\right )}-\frac {3 e \cos (2 d+2 e x) F^{c (a+b x)}}{16 \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac {3 e \cos (6 d+6 e x) F^{c (a+b x)}}{16 \left (b^2 c^2 \log ^2(F)+36 e^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4972

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]
 

Maple [A] (verified)

Time = 4.56 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.27

method	result	size
parallelrisch	\(\frac {6 \left (-\frac {\ln \left (F \right ) \sin \left (6 e x +6 d \right ) b c \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}{6}+\left (b^{2} c^{2} \ln \left (F \right )^{2} e +4 e^{3}\right ) \cos \left (6 e x +6 d \right )+\frac {\left (b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}\right ) \left (b c \ln \left (F \right ) \sin \left (2 e x +2 d \right )-2 e \cos \left (2 e x +2 d \right )\right )}{2}\right ) F^{c \left (b x +a \right )}}{32 b^{4} c^{4} \ln \left (F \right )^{4}+1280 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4608 e^{4}}\)	\(154\)
risch	\(\frac {3 e \,F^{c \left (b x +a \right )} \cos \left (6 e x +6 d \right )}{16 \left (b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}\right )}-\frac {c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (6 e x +6 d \right )}{32 \left (b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}\right )}-\frac {3 e \,F^{c \left (b x +a \right )} \cos \left (2 e x +2 d \right )}{16 \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {3 c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (2 e x +2 d \right )}{32 \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}\)	\(168\)
default	\(-\frac {F^{a c} \left (\frac {\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{2}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {8 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan \left (e x +d \right )^{2}}+\frac {-\frac {8 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+12 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}+\frac {8 e \left (11 b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{2}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}-\frac {8 e \left (11 b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{4}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}+\frac {8 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+12 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{6}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}+\frac {8 b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}+12 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}+\frac {8 b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}+12 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{5}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}-\frac {16 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{3}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}}{\left (1+\tan \left (e x +d \right )^{2}\right )^{3}}\right )}{32}\)	\(625\)
orering	\(\frac {4 b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}+20 e^{2}\right ) F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}+20 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3}-3 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{4} e +3 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{4} \sin \left (e x +d \right )^{2} e \right )}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}+\frac {4 b c \ln \left (F \right ) \left (F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3}-6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{4} e +6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{4} \sin \left (e x +d \right )^{2} e +6 F^{c \left (b x +a \right )} \cos \left (e x +d \right ) \sin \left (e x +d \right )^{5} e^{2}-24 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3} e^{2}+6 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{5} \sin \left (e x +d \right ) e^{2}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3}-9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{4} e +9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{4} \sin \left (e x +d \right )^{2} e +18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right ) \sin \left (e x +d \right )^{5} e^{2}-72 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3} e^{2}+18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{5} \sin \left (e x +d \right ) e^{2}-6 F^{c \left (b x +a \right )} e^{3} \sin \left (e x +d \right )^{6}+102 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{4} e^{3}-102 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{4} \sin \left (e x +d \right )^{2} e^{3}+6 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{6} e^{3}}{b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+144 e^{4}}\)	\(803\)

Input:

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

Output:

6*(-1/6*ln(F)*sin(6*e*x+6*d)*b*c*(4*e^2+b^2*c^2*ln(F)^2)+(b^2*c^2*ln(F)^2* 
e+4*e^3)*cos(6*e*x+6*d)+1/2*(b^2*c^2*ln(F)^2+36*e^2)*(b*c*ln(F)*sin(2*e*x+ 
2*d)-2*e*cos(2*e*x+2*d)))*F^(c*(b*x+a))/(32*b^4*c^4*ln(F)^4+1280*b^2*c^2*e 
^2*ln(F)^2+4608*e^4)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.88 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=\frac {{\left (24 \, e^{3} \cos \left (e x + d\right )^{6} - 36 \, e^{3} \cos \left (e x + d\right )^{4} + 6 \, e^{3} + 3 \, {\left (2 \, b^{2} c^{2} e \cos \left (e x + d\right )^{6} - 3 \, b^{2} c^{2} e \cos \left (e x + d\right )^{4} + b^{2} c^{2} e \cos \left (e x + d\right )^{2}\right )} \log \left (F\right )^{2} - {\left ({\left (b^{3} c^{3} \cos \left (e x + d\right )^{5} - b^{3} c^{3} \cos \left (e x + d\right )^{3}\right )} \log \left (F\right )^{3} + 2 \, {\left (2 \, b c e^{2} \cos \left (e x + d\right )^{5} - 2 \, b c e^{2} \cos \left (e x + d\right )^{3} - 3 \, b c e^{2} \cos \left (e x + d\right )\right )} \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 40 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 144 \, e^{4}} \] Input:

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

Output:

(24*e^3*cos(e*x + d)^6 - 36*e^3*cos(e*x + d)^4 + 6*e^3 + 3*(2*b^2*c^2*e*co 
s(e*x + d)^6 - 3*b^2*c^2*e*cos(e*x + d)^4 + b^2*c^2*e*cos(e*x + d)^2)*log( 
F)^2 - ((b^3*c^3*cos(e*x + d)^5 - b^3*c^3*cos(e*x + d)^3)*log(F)^3 + 2*(2* 
b*c*e^2*cos(e*x + d)^5 - 2*b*c*e^2*cos(e*x + d)^3 - 3*b*c*e^2*cos(e*x + d) 
)*log(F))*sin(e*x + d))*F^(b*c*x + a*c)/(b^4*c^4*log(F)^4 + 40*b^2*c^2*e^2 
*log(F)^2 + 144*e^4)
 

Sympy [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (115) = 230\).

Time = 0.08 (sec) , antiderivative size = 823, normalized size of antiderivative = 6.80 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/64*((F^(a*c)*b^3*c^3*log(F)^3*sin(6*d) - 6*F^(a*c)*b^2*c^2*e*cos(6*d)*l 
og(F)^2 + 4*F^(a*c)*b*c*e^2*log(F)*sin(6*d) - 24*F^(a*c)*e^3*cos(6*d))*F^( 
b*c*x)*cos(6*e*x) - (F^(a*c)*b^3*c^3*log(F)^3*sin(6*d) + 6*F^(a*c)*b^2*c^2 
*e*cos(6*d)*log(F)^2 + 4*F^(a*c)*b*c*e^2*log(F)*sin(6*d) + 24*F^(a*c)*e^3* 
cos(6*d))*F^(b*c*x)*cos(6*e*x + 12*d) + 3*(F^(a*c)*b^3*c^3*log(F)^3*sin(6* 
d) + 2*F^(a*c)*b^2*c^2*e*cos(6*d)*log(F)^2 + 36*F^(a*c)*b*c*e^2*log(F)*sin 
(6*d) + 72*F^(a*c)*e^3*cos(6*d))*F^(b*c*x)*cos(2*e*x + 8*d) - 3*(F^(a*c)*b 
^3*c^3*log(F)^3*sin(6*d) - 2*F^(a*c)*b^2*c^2*e*cos(6*d)*log(F)^2 + 36*F^(a 
*c)*b*c*e^2*log(F)*sin(6*d) - 72*F^(a*c)*e^3*cos(6*d))*F^(b*c*x)*cos(2*e*x 
 - 4*d) + (F^(a*c)*b^3*c^3*cos(6*d)*log(F)^3 + 6*F^(a*c)*b^2*c^2*e*log(F)^ 
2*sin(6*d) + 4*F^(a*c)*b*c*e^2*cos(6*d)*log(F) + 24*F^(a*c)*e^3*sin(6*d))* 
F^(b*c*x)*sin(6*e*x) + (F^(a*c)*b^3*c^3*cos(6*d)*log(F)^3 - 6*F^(a*c)*b^2* 
c^2*e*log(F)^2*sin(6*d) + 4*F^(a*c)*b*c*e^2*cos(6*d)*log(F) - 24*F^(a*c)*e 
^3*sin(6*d))*F^(b*c*x)*sin(6*e*x + 12*d) - 3*(F^(a*c)*b^3*c^3*cos(6*d)*log 
(F)^3 - 2*F^(a*c)*b^2*c^2*e*log(F)^2*sin(6*d) + 36*F^(a*c)*b*c*e^2*cos(6*d 
)*log(F) - 72*F^(a*c)*e^3*sin(6*d))*F^(b*c*x)*sin(2*e*x + 8*d) - 3*(F^(a*c 
)*b^3*c^3*cos(6*d)*log(F)^3 + 2*F^(a*c)*b^2*c^2*e*log(F)^2*sin(6*d) + 36*F 
^(a*c)*b*c*e^2*cos(6*d)*log(F) + 72*F^(a*c)*e^3*sin(6*d))*F^(b*c*x)*sin(2* 
e*x - 4*d))/(b^4*c^4*cos(6*d)^2*log(F)^4 + b^4*c^4*log(F)^4*sin(6*d)^2 + 1 
44*(cos(6*d)^2 + sin(6*d)^2)*e^4 + 40*(b^2*c^2*cos(6*d)^2*log(F)^2 + b^...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 1281, normalized size of antiderivative = 10.59 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*(2*b*c*log(abs(F))*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a 
*c*sgn(F) - 1/2*pi*a*c + 6*e*x + 6*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*s 
gn(F) - pi*b*c + 12*e)^2) - (pi*b*c*sgn(F) - pi*b*c + 12*e)*cos(1/2*pi*b*c 
*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + 6*e*x + 6*d)/( 
4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 12*e)^2))*e^(b*c*x*log 
(abs(F)) + a*c*log(abs(F))) + 3/32*(2*b*c*log(abs(F))*sin(1/2*pi*b*c*x*sgn 
(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + 2*e*x + 2*d)/(4*b^2* 
c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 4*e)^2) - (pi*b*c*sgn(F) - p 
i*b*c + 4*e)*cos(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 
1/2*pi*a*c + 2*e*x + 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b 
*c + 4*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 3/32*(2*b*c*log(ab 
s(F))*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi* 
a*c - 2*e*x - 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 4* 
e)^2) - (pi*b*c*sgn(F) - pi*b*c - 4*e)*cos(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b* 
c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - 2*e*x - 2*d)/(4*b^2*c^2*log(abs(F)) 
^2 + (pi*b*c*sgn(F) - pi*b*c - 4*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs 
(F))) + 1/32*(2*b*c*log(abs(F))*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1 
/2*pi*a*c*sgn(F) - 1/2*pi*a*c - 6*e*x - 6*d)/(4*b^2*c^2*log(abs(F))^2 + (p 
i*b*c*sgn(F) - pi*b*c - 12*e)^2) - (pi*b*c*sgn(F) - pi*b*c - 12*e)*cos(1/2 
*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - 6*e*...
 

Mupad [B] (verification not implemented)

Time = 16.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.67 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (2\,e\,x\right )+\sin \left (2\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,d\right )+\sin \left (2\,d\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,\left (b\,c\,\ln \left (F\right )+e\,2{}\mathrm {i}\right )}-\frac {3\,F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (2\,e\,x\right )-\sin \left (2\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,d\right )-\sin \left (2\,d\right )\,1{}\mathrm {i}\right )}{64\,\left (2\,e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}+\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (6\,e\,x\right )+\sin \left (6\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,d\right )+\sin \left (6\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,\left (b\,c\,\ln \left (F\right )+e\,6{}\mathrm {i}\right )}+\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (6\,e\,x\right )-\sin \left (6\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,d\right )-\sin \left (6\,d\right )\,1{}\mathrm {i}\right )}{64\,\left (6\,e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )} \] Input:

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

Output:

(F^(c*(a + b*x))*(cos(6*e*x) + sin(6*e*x)*1i)*(cos(6*d) + sin(6*d)*1i)*1i) 
/(64*(e*6i + b*c*log(F))) - (3*F^(c*(a + b*x))*(cos(2*e*x) - sin(2*e*x)*1i 
)*(cos(2*d) - sin(2*d)*1i))/(64*(2*e + b*c*log(F)*1i)) - (F^(c*(a + b*x))* 
(cos(2*e*x) + sin(2*e*x)*1i)*(cos(2*d) + sin(2*d)*1i)*3i)/(64*(e*2i + b*c* 
log(F))) + (F^(c*(a + b*x))*(cos(6*e*x) - sin(6*e*x)*1i)*(cos(6*d) - sin(6 
*d)*1i))/(64*(6*e + b*c*log(F)*1i))
 

Reduce [F]

\[ \int F^{c (a+b x)} \cos ^3(d+e x) \sin ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3}d x \right ) \] Input:

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

Output:

f**(a*c)*int(f**(b*c*x)*cos(d + e*x)**3*sin(d + e*x)**3,x)