\(\int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx\) [22]

Optimal result

Integrand size = 29, antiderivative size = 199 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=-\frac {e^{-3 i (d+e x)} F^{c (a+b x)} (i f+g)^3}{8 (3 i e-b c \log (F))}-\frac {3 e^{i (d+e x)} F^{c (a+b x)} (f-i g) (f+i g)^2}{8 (e-i b c \log (F))}+\frac {e^{3 i (d+e x)} F^{c (a+b x)} (f+i g)^3}{8 (3 e-i b c \log (F))}-\frac {3 e^{-i (d+e x)} F^{c (a+b x)} (f-i g)^2 (f+i g)}{8 (e+i b c \log (F))} \] Output:

-1/8*F^(c*(b*x+a))*(I*f+g)^3/exp(3*I*(e*x+d))/(3*I*e-b*c*ln(F))-3*exp(I*(e 
*x+d))*F^(c*(b*x+a))*(f-I*g)*(f+I*g)^2/(8*e-8*I*b*c*ln(F))+exp(3*I*(e*x+d) 
)*F^(c*(b*x+a))*(f+I*g)^3/(24*e-8*I*b*c*ln(F))-3/8*F^(c*(b*x+a))*(f-I*g)^2 
*(f+I*g)/exp(I*(e*x+d))/(e+I*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.15 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=\frac {1}{4} F^{c (a+b x)} \left (-\frac {3 (f-i g) (f+i g) \cos (d+e x) (e f-b c g \log (F))}{e^2+b^2 c^2 \log ^2(F)}+\frac {\cos (3 (d+e x)) \left (3 e f \left (f^2-3 g^2\right )+b c g \left (-3 f^2+g^2\right ) \log (F)\right )}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {3 (f-i g) (f+i g) (e g+b c f \log (F)) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac {\left (e \left (9 f^2 g-3 g^3\right )+b c f \left (f^2-3 g^2\right ) \log (F)\right ) \sin (3 (d+e x))}{9 e^2+b^2 c^2 \log ^2(F)}\right ) \] Input:

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

Output:

(F^(c*(a + b*x))*((-3*(f - I*g)*(f + I*g)*Cos[d + e*x]*(e*f - b*c*g*Log[F] 
))/(e^2 + b^2*c^2*Log[F]^2) + (Cos[3*(d + e*x)]*(3*e*f*(f^2 - 3*g^2) + b*c 
*g*(-3*f^2 + g^2)*Log[F]))/(9*e^2 + b^2*c^2*Log[F]^2) + (3*(f - I*g)*(f + 
I*g)*(e*g + b*c*f*Log[F])*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - ((e*(9* 
f^2*g - 3*g^3) + b*c*f*(f^2 - 3*g^2)*Log[F])*Sin[3*(d + e*x)])/(9*e^2 + b^ 
2*c^2*Log[F]^2)))/4
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(797\) vs. \(2(199)=398\).

Time = 1.26 (sec) , antiderivative size = 797, normalized size of antiderivative = 4.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {7292, 7293, 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (f^3 \sin ^3(d+e x) F^{a c+b c x}+3 f^2 g \sin ^2(d+e x) \cos (d+e x) F^{a c+b c x}+3 f g^2 \sin (d+e x) \cos ^2(d+e x) F^{a c+b c x}+g^3 \cos ^3(d+e x) F^{a c+b c x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b c f^3 \log (F) \sin ^3(d+e x) F^{a c+b x c}}{9 e^2+b^2 c^2 \log ^2(F)}-\frac {3 e f^3 \cos (d+e x) \sin ^2(d+e x) F^{a c+b x c}}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {3 e f^2 g \sin (d+e x) F^{a c+b x c}}{4 \left (e^2+b^2 c^2 \log ^2(F)\right )}+\frac {3 b c f g^2 \log (F) \sin (d+e x) F^{a c+b x c}}{4 \left (e^2+b^2 c^2 \log ^2(F)\right )}+\frac {3 e g^3 \cos ^2(d+e x) \sin (d+e x) F^{a c+b x c}}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 g^3 \sin (d+e x) F^{a c+b x c}}{9 e^4+10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 f^3 \log (F) \sin (d+e x) F^{a c+b x c}}{9 e^4+10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}-\frac {9 e f^2 g \sin (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {3 b c f g^2 \log (F) \sin (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2+b^2 c^2 \log ^2(F)\right )}-\frac {3 e f g^2 \cos (d+e x) F^{a c+b x c}}{4 \left (e^2+b^2 c^2 \log ^2(F)\right )}+\frac {3 b c f^2 g \cos (d+e x) \log (F) F^{a c+b x c}}{4 \left (e^2+b^2 c^2 \log ^2(F)\right )}-\frac {9 e f g^2 \cos (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {b c g^3 \cos ^3(d+e x) \log (F) F^{a c+b x c}}{9 e^2+b^2 c^2 \log ^2(F)}-\frac {3 b c f^2 g \cos (3 d+3 e x) \log (F) F^{a c+b x c}}{4 \left (9 e^2+b^2 c^2 \log ^2(F)\right )}-\frac {6 e^3 f^3 \cos (d+e x) F^{a c+b x c}}{9 e^4+10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 g^3 \cos (d+e x) \log (F) F^{a c+b x c}}{9 e^4+10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}\)

Input:

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

Output:

(-3*e*f*F^(a*c + b*c*x)*g^2*Cos[d + e*x])/(4*(e^2 + b^2*c^2*Log[F]^2)) + ( 
3*b*c*f^2*F^(a*c + b*c*x)*g*Cos[d + e*x]*Log[F])/(4*(e^2 + b^2*c^2*Log[F]^ 
2)) - (9*e*f*F^(a*c + b*c*x)*g^2*Cos[3*d + 3*e*x])/(4*(9*e^2 + b^2*c^2*Log 
[F]^2)) + (b*c*F^(a*c + b*c*x)*g^3*Cos[d + e*x]^3*Log[F])/(9*e^2 + b^2*c^2 
*Log[F]^2) - (3*b*c*f^2*F^(a*c + b*c*x)*g*Cos[3*d + 3*e*x]*Log[F])/(4*(9*e 
^2 + b^2*c^2*Log[F]^2)) - (6*e^3*f^3*F^(a*c + b*c*x)*Cos[d + e*x])/(9*e^4 
+ 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*F^(a*c + b*c*x) 
*g^3*Cos[d + e*x]*Log[F])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F 
]^4) + (3*e*f^2*F^(a*c + b*c*x)*g*Sin[d + e*x])/(4*(e^2 + b^2*c^2*Log[F]^2 
)) + (3*b*c*f*F^(a*c + b*c*x)*g^2*Log[F]*Sin[d + e*x])/(4*(e^2 + b^2*c^2*L 
og[F]^2)) + (3*e*F^(a*c + b*c*x)*g^3*Cos[d + e*x]^2*Sin[d + e*x])/(9*e^2 + 
 b^2*c^2*Log[F]^2) + (6*e^3*F^(a*c + b*c*x)*g^3*Sin[d + e*x])/(9*e^4 + 10* 
b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*f^3*F^(a*c + b*c*x)* 
Log[F]*Sin[d + e*x])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) 
- (3*e*f^3*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x]^2)/(9*e^2 + b^2*c^2*L 
og[F]^2) + (b*c*f^3*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x]^3)/(9*e^2 + b^2*c^ 
2*Log[F]^2) - (9*e*f^2*F^(a*c + b*c*x)*g*Sin[3*d + 3*e*x])/(4*(9*e^2 + b^2 
*c^2*Log[F]^2)) + (3*b*c*f*F^(a*c + b*c*x)*g^2*Log[F]*Sin[3*d + 3*e*x])/(4 
*(9*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 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 4.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {\left (i f^{3}-3 i f \,g^{2}-3 f^{2} g +g^{3}\right ) F^{c \left (b x +a \right )} {\mathrm e}^{3 i \left (e x +d \right )}}{24 i e +8 b c \ln \left (F \right )}+\frac {3 \left (-i f^{3}-i f \,g^{2}+f^{2} g +g^{3}\right ) F^{c \left (b x +a \right )} {\mathrm e}^{i \left (e x +d \right )}}{8 \left (i e +b c \ln \left (F \right )\right )}+\frac {3 \left (i f^{3}+i f \,g^{2}+f^{2} g +g^{3}\right ) F^{c \left (b x +a \right )} {\mathrm e}^{-i \left (e x +d \right )}}{8 \left (b c \ln \left (F \right )-i e \right )}+\frac {\left (-i f^{3}+3 i f \,g^{2}-3 f^{2} g +g^{3}\right ) F^{c \left (b x +a \right )} {\mathrm e}^{-3 i \left (e x +d \right )}}{8 b c \ln \left (F \right )-24 i e}\)	\(220\)
parallelrisch	\(\frac {3 F^{c \left (b x +a \right )} \left (-\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (b c g \left (f^{2}-\frac {g^{2}}{3}\right ) \ln \left (F \right )-e f \left (f^{2}-3 g^{2}\right )\right ) \cos \left (3 e x +3 d \right )-\frac {\left (b c f \left (f^{2}-3 g^{2}\right ) \ln \left (F \right )+\left (9 f^{2} g -3 g^{3}\right ) e \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \sin \left (3 e x +3 d \right )}{3}+\left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (\left (b c g \ln \left (F \right )-e f \right ) \cos \left (e x +d \right )+\sin \left (e x +d \right ) \left (\ln \left (F \right ) b c f +e g \right )\right ) \left (f^{2}+g^{2}\right )\right )}{4 b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+36 e^{4}}\)	\(220\)
orering	\(\frac {4 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{3}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{3}+3 F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{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} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{3}+6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )+6 F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right ) \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )^{2}+3 F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (-g \,e^{2} \cos \left (e x +d \right )-f \,e^{2} \sin \left (e x +d \right )\right )\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{3}+9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )+18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right ) \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )^{2}+9 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (-g \,e^{2} \cos \left (e x +d \right )-f \,e^{2} \sin \left (e x +d \right )\right )+6 F^{c \left (b x +a \right )} \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right )^{3}+18 F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right ) \left (-g e \sin \left (e x +d \right )+f e \cos \left (e x +d \right )\right ) \left (-g \,e^{2} \cos \left (e x +d \right )-f \,e^{2} \sin \left (e x +d \right )\right )+3 F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{2} \left (\sin \left (e x +d \right ) e^{3} g -f \,e^{3} \cos \left (e x +d \right )\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}\)	\(873\)
norman	\(\text {Expression too large to display}\)	\(1032\)
parts	\(\text {Expression too large to display}\)	\(2438\)
default	\(\text {Expression too large to display}\)	\(2451\)

Input:

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

Output:

1/8*(I*f^3-3*I*f*g^2-3*f^2*g+g^3)/(3*I*e+b*c*ln(F))*F^(c*(b*x+a))*exp(3*I* 
(e*x+d))+3/8*(-I*f^3-I*f*g^2+f^2*g+g^3)/(I*e+b*c*ln(F))*F^(c*(b*x+a))*exp( 
I*(e*x+d))+3/8*(f^2*g+g^3+I*f^3+I*f*g^2)/(b*c*ln(F)-I*e)*F^(c*(b*x+a))*exp 
(-I*(e*x+d))+1/8*(-I*f^3+3*I*f*g^2-3*f^2*g+g^3)/(b*c*ln(F)-3*I*e)*F^(c*(b* 
x+a))*exp(-3*I*(e*x+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (163) = 326\).

Time = 0.41 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.51 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=-\frac {{\left (9 \, e^{3} f^{3} \cos \left (e x + d\right ) - 3 \, {\left (e^{3} f^{3} - 3 \, e^{3} f g^{2}\right )} \cos \left (e x + d\right )^{3} - {\left (3 \, b^{3} c^{3} f^{2} g \cos \left (e x + d\right ) - {\left (3 \, b^{3} c^{3} f^{2} g - b^{3} c^{3} g^{3}\right )} \cos \left (e x + d\right )^{3}\right )} \log \left (F\right )^{3} - 3 \, {\left ({\left (b^{2} c^{2} e f^{3} - 3 \, b^{2} c^{2} e f g^{2}\right )} \cos \left (e x + d\right )^{3} - {\left (b^{2} c^{2} e f^{3} - 2 \, b^{2} c^{2} e f g^{2}\right )} \cos \left (e x + d\right )\right )} \log \left (F\right )^{2} + {\left ({\left (3 \, b c e^{2} f^{2} g - b c e^{2} g^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (3 \, b c e^{2} f^{2} g + 2 \, b c e^{2} g^{3}\right )} \cos \left (e x + d\right )\right )} \log \left (F\right ) - {\left (9 \, e^{3} f^{2} g + 6 \, e^{3} g^{3} + {\left (b^{3} c^{3} f^{3} - {\left (b^{3} c^{3} f^{3} - 3 \, b^{3} c^{3} f g^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \log \left (F\right )^{3} - 3 \, {\left (3 \, e^{3} f^{2} g - e^{3} g^{3}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (b^{2} c^{2} e f^{2} g - {\left (3 \, b^{2} c^{2} e f^{2} g - b^{2} c^{2} e g^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \log \left (F\right )^{2} + {\left (7 \, b c e^{2} f^{3} + 6 \, b c e^{2} f g^{2} - {\left (b c e^{2} f^{3} - 3 \, b c e^{2} f g^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 10 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 9 \, e^{4}} \] Input:

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

Output:

-(9*e^3*f^3*cos(e*x + d) - 3*(e^3*f^3 - 3*e^3*f*g^2)*cos(e*x + d)^3 - (3*b 
^3*c^3*f^2*g*cos(e*x + d) - (3*b^3*c^3*f^2*g - b^3*c^3*g^3)*cos(e*x + d)^3 
)*log(F)^3 - 3*((b^2*c^2*e*f^3 - 3*b^2*c^2*e*f*g^2)*cos(e*x + d)^3 - (b^2* 
c^2*e*f^3 - 2*b^2*c^2*e*f*g^2)*cos(e*x + d))*log(F)^2 + ((3*b*c*e^2*f^2*g 
- b*c*e^2*g^3)*cos(e*x + d)^3 - 3*(3*b*c*e^2*f^2*g + 2*b*c*e^2*g^3)*cos(e* 
x + d))*log(F) - (9*e^3*f^2*g + 6*e^3*g^3 + (b^3*c^3*f^3 - (b^3*c^3*f^3 - 
3*b^3*c^3*f*g^2)*cos(e*x + d)^2)*log(F)^3 - 3*(3*e^3*f^2*g - e^3*g^3)*cos( 
e*x + d)^2 + 3*(b^2*c^2*e*f^2*g - (3*b^2*c^2*e*f^2*g - b^2*c^2*e*g^3)*cos( 
e*x + d)^2)*log(F)^2 + (7*b*c*e^2*f^3 + 6*b*c*e^2*f*g^2 - (b*c*e^2*f^3 - 3 
*b*c*e^2*f*g^2)*cos(e*x + d)^2)*log(F))*sin(e*x + d))*F^(b*c*x + a*c)/(b^4 
*c^4*log(F)^4 + 10*b^2*c^2*e^2*log(F)^2 + 9*e^4)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7009 vs. \(2 (163) = 326\).

Time = 4.06 (sec) , antiderivative size = 7009, normalized size of antiderivative = 35.22 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((x*(f*sin(d) + g*cos(d))**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*( 
f*sin(d) + g*cos(d))**3, Eq(b, 0) & Eq(e, 0)), (x*(f*sin(d) + g*cos(d))**3 
, Eq(c, 0) & Eq(e, 0)), (-3*F**(a*c + b*c*x)*f**3*x*sin(I*b*c*x*log(F) - d 
)**3/8 + 3*I*F**(a*c + b*c*x)*f**3*x*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c* 
x*log(F) - d)/8 - 3*F**(a*c + b*c*x)*f**3*x*sin(I*b*c*x*log(F) - d)*cos(I* 
b*c*x*log(F) - d)**2/8 + 3*I*F**(a*c + b*c*x)*f**3*x*cos(I*b*c*x*log(F) - 
d)**3/8 + 3*I*F**(a*c + b*c*x)*f**2*g*x*sin(I*b*c*x*log(F) - d)**3/8 + 3*F 
**(a*c + b*c*x)*f**2*g*x*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d 
)/8 + 3*I*F**(a*c + b*c*x)*f**2*g*x*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*lo 
g(F) - d)**2/8 + 3*F**(a*c + b*c*x)*f**2*g*x*cos(I*b*c*x*log(F) - d)**3/8 
- 3*F**(a*c + b*c*x)*f*g**2*x*sin(I*b*c*x*log(F) - d)**3/8 + 3*I*F**(a*c + 
 b*c*x)*f*g**2*x*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/8 - 3* 
F**(a*c + b*c*x)*f*g**2*x*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)* 
*2/8 + 3*I*F**(a*c + b*c*x)*f*g**2*x*cos(I*b*c*x*log(F) - d)**3/8 + 3*I*F* 
*(a*c + b*c*x)*g**3*x*sin(I*b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*g* 
*3*x*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/8 + 3*I*F**(a*c + 
b*c*x)*g**3*x*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)**2/8 + 3*F** 
(a*c + b*c*x)*g**3*x*cos(I*b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*f** 
3*sin(I*b*c*x*log(F) - d)**3/(8*b*c*log(F)) - I*F**(a*c + b*c*x)*f**3*sin( 
I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/(b*c*log(F)) + F**(a*c +...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3263 vs. \(2 (163) = 326\).

Time = 0.21 (sec) , antiderivative size = 3263, normalized size of antiderivative = 16.40 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*((F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) - 3*F^(a*c)*b^2*c^2*e*cos(3*d)*lo 
g(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(3*d) - 3*F^(a*c)*e^3*cos(3*d))*F^(b*c* 
x)*cos(3*e*x) - (F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) + 3*F^(a*c)*b^2*c^2*e*c 
os(3*d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(3*d) + 3*F^(a*c)*e^3*cos(3*d 
))*F^(b*c*x)*cos(3*e*x + 6*d) + 3*(F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) + F^( 
a*c)*b^2*c^2*e*cos(3*d)*log(F)^2 + 9*F^(a*c)*b*c*e^2*log(F)*sin(3*d) + 9*F 
^(a*c)*e^3*cos(3*d))*F^(b*c*x)*cos(e*x + 4*d) - 3*(F^(a*c)*b^3*c^3*log(F)^ 
3*sin(3*d) - F^(a*c)*b^2*c^2*e*cos(3*d)*log(F)^2 + 9*F^(a*c)*b*c*e^2*log(F 
)*sin(3*d) - 9*F^(a*c)*e^3*cos(3*d))*F^(b*c*x)*cos(e*x - 2*d) + (F^(a*c)*b 
^3*c^3*cos(3*d)*log(F)^3 + 3*F^(a*c)*b^2*c^2*e*log(F)^2*sin(3*d) + F^(a*c) 
*b*c*e^2*cos(3*d)*log(F) + 3*F^(a*c)*e^3*sin(3*d))*F^(b*c*x)*sin(3*e*x) + 
(F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 - 3*F^(a*c)*b^2*c^2*e*log(F)^2*sin(3*d) 
 + F^(a*c)*b*c*e^2*cos(3*d)*log(F) - 3*F^(a*c)*e^3*sin(3*d))*F^(b*c*x)*sin 
(3*e*x + 6*d) - 3*(F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 - F^(a*c)*b^2*c^2*e*l 
og(F)^2*sin(3*d) + 9*F^(a*c)*b*c*e^2*cos(3*d)*log(F) - 9*F^(a*c)*e^3*sin(3 
*d))*F^(b*c*x)*sin(e*x + 4*d) - 3*(F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 + F^( 
a*c)*b^2*c^2*e*log(F)^2*sin(3*d) + 9*F^(a*c)*b*c*e^2*cos(3*d)*log(F) + 9*F 
^(a*c)*e^3*sin(3*d))*F^(b*c*x)*sin(e*x - 2*d))*f^3/(b^4*c^4*cos(3*d)^2*log 
(F)^4 + b^4*c^4*log(F)^4*sin(3*d)^2 + 9*(cos(3*d)^2 + sin(3*d)^2)*e^4 + 10 
*(b^2*c^2*cos(3*d)^2*log(F)^2 + b^2*c^2*log(F)^2*sin(3*d)^2)*e^2) - 3/8...
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1473 vs. \(2 (163) = 326\).

Time = 0.31 (sec) , antiderivative size = 1473, normalized size of antiderivative = 7.40 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

I*((f^3 - 3*I*f^2*g - 3*f*g^2 + I*g^3)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi 
*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + 3*I*e*x + 3*I*d)/(8*I*pi*b*c 
*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*I*e) - (f^3 - 3*I*f^2*g - 3 
*f*g^2 + I*g^3)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c* 
sgn(F) + 1/2*I*pi*a*c - 3*I*e*x - 3*I*d)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c 
+ 16*b*c*log(abs(F)) - 48*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 
((I*f^3 + 3*f^2*g - 3*I*f*g^2 - g^3)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b 
*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + 3*I*e*x + 3*I*d)/(8*I*pi*b*c*s 
gn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*I*e) + (I*f^3 + 3*f^2*g - 3*I 
*f*g^2 - g^3)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sg 
n(F) + 1/2*I*pi*a*c - 3*I*e*x - 3*I*d)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 
16*b*c*log(abs(F)) - 48*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 3* 
I*((f^3 - I*f^2*g + f*g^2 - I*g^3)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c 
*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + I*e*x + I*d)/(8*I*pi*b*c*sgn(F) 
- 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*I*e) - (f^3 - I*f^2*g + f*g^2 - I*g 
^3)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2 
*I*pi*a*c - I*e*x - I*d)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs 
(F)) - 16*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 3*((-I*f^3 - f^2 
*g - I*f*g^2 - g^3)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a 
*c*sgn(F) - 1/2*I*pi*a*c + I*e*x + I*d)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c...
 

Mupad [B] (verification not implemented)

Time = 19.53 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx=\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {e}}^{-d\,3{}\mathrm {i}-e\,x\,3{}\mathrm {i}}\,{\left (g+f\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{24\,e+b\,c\,\ln \left (F\right )\,8{}\mathrm {i}}+\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {e}}^{d\,3{}\mathrm {i}+e\,x\,3{}\mathrm {i}}\,{\left (f+g\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{8\,b\,c\,\ln \left (F\right )+e\,24{}\mathrm {i}}+\frac {3\,F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {e}}^{-d\,1{}\mathrm {i}-e\,x\,1{}\mathrm {i}}\,\left (f+g\,1{}\mathrm {i}\right )\,{\left (g+f\,1{}\mathrm {i}\right )}^2}{8\,e+b\,c\,\ln \left (F\right )\,8{}\mathrm {i}}-\frac {3\,F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {e}}^{d\,1{}\mathrm {i}+e\,x\,1{}\mathrm {i}}\,{\left (f+g\,1{}\mathrm {i}\right )}^2\,\left (g+f\,1{}\mathrm {i}\right )}{8\,b\,c\,\ln \left (F\right )+e\,8{}\mathrm {i}} \] Input:

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

Output:

(F^(b*c*x)*F^(a*c)*exp(- d*3i - e*x*3i)*(f*1i + g)^3*1i)/(24*e + b*c*log(F 
)*8i) + (F^(b*c*x)*F^(a*c)*exp(d*3i + e*x*3i)*(f + g*1i)^3*1i)/(e*24i + 8* 
b*c*log(F)) + (3*F^(b*c*x)*F^(a*c)*exp(- d*1i - e*x*1i)*(f + g*1i)*(f*1i + 
 g)^2)/(8*e + b*c*log(F)*8i) - (3*F^(b*c*x)*F^(a*c)*exp(d*1i + e*x*1i)*(f 
+ g*1i)^2*(f*1i + g))/(e*8i + 8*b*c*log(F))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.12 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(f**(a*c + b*c*x)*(cos(d + e*x)**3*log(f)**3*b**3*c**3*g**3 - 3*cos(d + e* 
x)**3*log(f)**2*b**2*c**2*e*f*g**2 + 6*cos(d + e*x)**3*log(f)*b*c*e**2*f** 
2*g + 7*cos(d + e*x)**3*log(f)*b*c*e**2*g**3 - 6*cos(d + e*x)**3*e**3*f**3 
 - 9*cos(d + e*x)**3*e**3*f*g**2 + 3*cos(d + e*x)**2*log(f)**3*sin(d + e*x 
)*b**3*c**3*f*g**2 - 6*cos(d + e*x)**2*log(f)**2*sin(d + e*x)*b**2*c**2*e* 
f**2*g + 3*cos(d + e*x)**2*log(f)**2*sin(d + e*x)*b**2*c**2*e*g**3 + 6*cos 
(d + e*x)**2*log(f)*sin(d + e*x)*b*c*e**2*f**3 + 9*cos(d + e*x)**2*log(f)* 
sin(d + e*x)*b*c*e**2*f*g**2 + 9*cos(d + e*x)**2*sin(d + e*x)*e**3*g**3 + 
3*cos(d + e*x)*log(f)**3*sin(d + e*x)**2*b**3*c**3*f**2*g - 3*cos(d + e*x) 
*log(f)**2*sin(d + e*x)**2*b**2*c**2*e*f**3 + 6*cos(d + e*x)*log(f)**2*sin 
(d + e*x)**2*b**2*c**2*e*f*g**2 + 9*cos(d + e*x)*log(f)*sin(d + e*x)**2*b* 
c*e**2*f**2*g + 6*cos(d + e*x)*log(f)*sin(d + e*x)**2*b*c*e**2*g**3 - 9*co 
s(d + e*x)*sin(d + e*x)**2*e**3*f**3 + log(f)**3*sin(d + e*x)**3*b**3*c**3 
*f**3 + 3*log(f)**2*sin(d + e*x)**3*b**2*c**2*e*f**2*g + 7*log(f)*sin(d + 
e*x)**3*b*c*e**2*f**3 + 6*log(f)*sin(d + e*x)**3*b*c*e**2*f*g**2 + 9*sin(d 
 + e*x)**3*e**3*f**2*g + 6*sin(d + e*x)**3*e**3*g**3))/(log(f)**4*b**4*c** 
4 + 10*log(f)**2*b**2*c**2*e**2 + 9*e**4)