\(\int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx\) [139]

Optimal result

Integrand size = 22, antiderivative size = 245 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)} \]

[Out]

f^2*F^(b*c*x+a*c)/b/c/ln(F)+2*b*c*f^2*F^(b*c*x+a*c)*cos(e*x+d)*ln(F)/(e^2+b^2*c^2*ln(F)^2)+2*e^2*f^2*F^(b*c*x+
a*c)/b/c/ln(F)/(4*e^2+b^2*c^2*ln(F)^2)+b*c*f^2*F^(b*c*x+a*c)*cos(e*x+d)^2*ln(F)/(4*e^2+b^2*c^2*ln(F)^2)+2*e*f^
2*F^(b*c*x+a*c)*sin(e*x+d)/(e^2+b^2*c^2*ln(F)^2)+2*e*f^2*F^(b*c*x+a*c)*cos(e*x+d)*sin(e*x+d)/(4*e^2+b^2*c^2*ln
(F)^2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6873, 12, 6874, 2225, 4518, 4520} \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {2 e f^2 \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {b c f^2 \log (F) \cos ^2(d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac {2 b c f^2 \log (F) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {2 e f^2 \sin (d+e x) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \]

[In]

Int[F^(c*(a + b*x))*(f + f*Cos[d + e*x])^2,x]

[Out]

(f^2*F^(a*c + b*c*x))/(b*c*Log[F]) + (2*b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F]^2)
+ (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) + (b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]^
2*Log[F])/(4*e^2 + b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (2*e*
f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^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 2225

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 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[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 4520

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int f^2 F^{a c+b c x} (1+\cos (d+e x))^2 \, dx \\ & = f^2 \int F^{a c+b c x} (1+\cos (d+e x))^2 \, dx \\ & = f^2 \int \left (F^{a c+b c x}+2 F^{a c+b c x} \cos (d+e x)+F^{a c+b c x} \cos ^2(d+e x)\right ) \, dx \\ & = f^2 \int F^{a c+b c x} \, dx+f^2 \int F^{a c+b c x} \cos ^2(d+e x) \, dx+\left (2 f^2\right ) \int F^{a c+b c x} \cos (d+e x) \, dx \\ & = \frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2 f^2\right ) \int F^{a c+b c x} \, dx}{4 e^2+b^2 c^2 \log ^2(F)} \\ & = \frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.93 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {f^2 F^{c (a+b x)} \left (12 e^4+15 b^2 c^2 e^2 \log ^2(F)+3 b^4 c^4 \log ^4(F)+b^2 c^2 \cos (2 (d+e x)) \log ^2(F) \left (e^2+b^2 c^2 \log ^2(F)\right )+4 b^2 c^2 \cos (d+e x) \log ^2(F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )+16 b c e^3 \log (F) \sin (d+e x)+4 b^3 c^3 e \log ^3(F) \sin (d+e x)+2 b c e^3 \log (F) \sin (2 (d+e x))+2 b^3 c^3 e \log ^3(F) \sin (2 (d+e x))\right )}{2 \left (4 b c e^4 \log (F)+5 b^3 c^3 e^2 \log ^3(F)+b^5 c^5 \log ^5(F)\right )} \]

[In]

Integrate[F^(c*(a + b*x))*(f + f*Cos[d + e*x])^2,x]

[Out]

(f^2*F^(c*(a + b*x))*(12*e^4 + 15*b^2*c^2*e^2*Log[F]^2 + 3*b^4*c^4*Log[F]^4 + b^2*c^2*Cos[2*(d + e*x)]*Log[F]^
2*(e^2 + b^2*c^2*Log[F]^2) + 4*b^2*c^2*Cos[d + e*x]*Log[F]^2*(4*e^2 + b^2*c^2*Log[F]^2) + 16*b*c*e^3*Log[F]*Si
n[d + e*x] + 4*b^3*c^3*e*Log[F]^3*Sin[d + e*x] + 2*b*c*e^3*Log[F]*Sin[2*(d + e*x)] + 2*b^3*c^3*e*Log[F]^3*Sin[
2*(d + e*x)]))/(2*(4*b*c*e^4*Log[F] + 5*b^3*c^3*e^2*Log[F]^3 + b^5*c^5*Log[F]^5))

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {3 f^{2} F^{c \left (x b +a \right )}}{2 b c \ln \left (F \right )}+\frac {2 \ln \left (F \right ) c b \,f^{2} F^{c \left (x b +a \right )} \cos \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {2 F^{c \left (x b +a \right )} e \,f^{2} \sin \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {\ln \left (F \right ) c b \,f^{2} F^{c \left (x b +a \right )} \cos \left (2 e x +2 d \right )}{2 b^{2} c^{2} \ln \left (F \right )^{2}+8 e^{2}}+\frac {e \,f^{2} F^{c \left (x b +a \right )} \sin \left (2 e x +2 d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}\)	\(193\)
parallelrisch	\(\frac {2 F^{c \left (x b +a \right )} f^{2} \left (\frac {b^{2} c^{2} \ln \left (F \right )^{2} \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \cos \left (2 e x +2 d \right )}{4}+\frac {c b e \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \sin \left (2 e x +2 d \right )}{2}+\left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (\cos \left (e x +d \right ) b^{2} c^{2} \ln \left (F \right )^{2}+\frac {3 b^{2} c^{2} \ln \left (F \right )^{2}}{4}+e \sin \left (e x +d \right ) \ln \left (F \right ) b c +\frac {3 e^{2}}{4}\right )\right )}{b c \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}\)	\(194\)
norman	\(\frac {\frac {12 e^{3} f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}}+\frac {4 \left (2 b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) e \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}}+\frac {2 f^{2} \left (2 b^{4} c^{4} \ln \left (F \right )^{4}+8 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+3 e^{4}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{b c \ln \left (F \right ) \left (b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}\right )}+\frac {6 e^{4} f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{b c \ln \left (F \right ) \left (b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}\right )}+\frac {12 e^{2} f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}\)	\(381\)
default	\(\frac {F^{a c} f^{2} \left (\frac {2 F^{b c x}}{b c \ln \left (F \right )}+\frac {\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}+\frac {\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {2 \left (b^{2} c^{2} \ln \left (F \right )^{2}+2 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}-\frac {4 \left (b^{2} c^{2} \ln \left (F \right )^{2}-2 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {2 \left (b^{2} c^{2} \ln \left (F \right )^{2}+2 e^{2}\right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}\right )}{2}\)	\(436\)
parts	\(\frac {f^{2} F^{c \left (x b +a \right )}}{b c \ln \left (F \right )}+\frac {\frac {\left (b^{2} c^{2} \ln \left (F \right )^{2}+2 e^{2}\right ) f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {\left (b^{2} c^{2} \ln \left (F \right )^{2}+2 e^{2}\right ) f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {4 e \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 e \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {2 f^{2} \left (b^{2} c^{2} \ln \left (F \right )^{2}-2 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}+\frac {\frac {4 e \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {2 \ln \left (F \right ) b c \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {2 \ln \left (F \right ) b c \,f^{2} {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}\)	\(477\)

[In]

int(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x,method=_RETURNVERBOSE)

[Out]

3/2/b/c/ln(F)*f^2*F^(c*(b*x+a))+2*ln(F)*c*b*f^2*F^(c*(b*x+a))/(e^2+b^2*c^2*ln(F)^2)*cos(e*x+d)+2*F^(c*(b*x+a))
*e*f^2/(e^2+b^2*c^2*ln(F)^2)*sin(e*x+d)+1/2/(4*e^2+b^2*c^2*ln(F)^2)*ln(F)*c*b*f^2*F^(c*(b*x+a))*cos(2*e*x+2*d)
+e*f^2*F^(c*(b*x+a))/(4*e^2+b^2*c^2*ln(F)^2)*sin(2*e*x+2*d)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.99 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {{\left (6 \, e^{4} f^{2} + {\left (b^{4} c^{4} f^{2} \cos \left (e x + d\right )^{2} + 2 \, b^{4} c^{4} f^{2} \cos \left (e x + d\right ) + b^{4} c^{4} f^{2}\right )} \log \left (F\right )^{4} + {\left (b^{2} c^{2} e^{2} f^{2} \cos \left (e x + d\right )^{2} + 8 \, b^{2} c^{2} e^{2} f^{2} \cos \left (e x + d\right ) + 7 \, b^{2} c^{2} e^{2} f^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left ({\left (b^{3} c^{3} e f^{2} \cos \left (e x + d\right ) + b^{3} c^{3} e f^{2}\right )} \log \left (F\right )^{3} + {\left (b c e^{3} f^{2} \cos \left (e x + d\right ) + 4 \, b c e^{3} f^{2}\right )} \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5} + 5 \, b^{3} c^{3} e^{2} \log \left (F\right )^{3} + 4 \, b c e^{4} \log \left (F\right )} \]

[In]

integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="fricas")

[Out]

(6*e^4*f^2 + (b^4*c^4*f^2*cos(e*x + d)^2 + 2*b^4*c^4*f^2*cos(e*x + d) + b^4*c^4*f^2)*log(F)^4 + (b^2*c^2*e^2*f
^2*cos(e*x + d)^2 + 8*b^2*c^2*e^2*f^2*cos(e*x + d) + 7*b^2*c^2*e^2*f^2)*log(F)^2 + 2*((b^3*c^3*e*f^2*cos(e*x +
d) + b^3*c^3*e*f^2)*log(F)^3 + (b*c*e^3*f^2*cos(e*x + d) + 4*b*c*e^3*f^2)*log(F))*sin(e*x + d))*F^(b*c*x + a*
c)/(b^5*c^5*log(F)^5 + 5*b^3*c^3*e^2*log(F)^3 + 4*b*c*e^4*log(F))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.97 (sec) , antiderivative size = 2394, normalized size of antiderivative = 9.77 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\text {Too large to display} \]

[In]

integrate(F**(c*(b*x+a))*(f+f*cos(e*x+d))**2,x)

[Out]

Piecewise((x*(f*cos(d) + f)**2, Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(e, 0)), (f**2*x*sin(d + e*x)**2/2 + f**2*x
*cos(d + e*x)**2/2 + f**2*x + f**2*sin(d + e*x)*cos(d + e*x)/(2*e) + 2*f**2*sin(d + e*x)/e, Eq(F, 1)), (F**(a*
c)*(f**2*x*sin(d + e*x)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x + f**2*sin(d + e*x)*cos(d + e*x)/(2*e) + 2*f*
*2*sin(d + e*x)/e), Eq(b, 0)), (f**2*x*sin(d + e*x)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x + f**2*sin(d + e*
x)*cos(d + e*x)/(2*e) + 2*f**2*sin(d + e*x)/e, Eq(c, 0)), (I*F**(a*c + b*c*x)*f**2*x*sin(I*b*c*x*log(F) - d) +
F**(a*c + b*c*x)*f**2*x*cos(I*b*c*x*log(F) - d) + 2*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F) - d)**2/(3*b*c*l
og(F)) - 2*I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)/(3*b*c*log(F)) - I*F**(a*c
+ b*c*x)*f**2*sin(I*b*c*x*log(F) - d)/(b*c*log(F)) + F**(a*c + b*c*x)*f**2*cos(I*b*c*x*log(F) - d)**2/(3*b*c*l
og(F)) + F**(a*c + b*c*x)*f**2/(b*c*log(F)), Eq(e, -I*b*c*log(F))), (-F**(a*c + b*c*x)*f**2*x*sin(I*b*c*x*log(
F)/2 - d)**2/4 + I*F**(a*c + b*c*x)*f**2*x*sin(I*b*c*x*log(F)/2 - d)*cos(I*b*c*x*log(F)/2 - d)/2 + F**(a*c + b
*c*x)*f**2*x*cos(I*b*c*x*log(F)/2 - d)**2/4 + I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F)/2 - d)*cos(I*b*c*x*lo
g(F)/2 - d)/(2*b*c*log(F)) + 4*I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F)/2 - d)/(3*b*c*log(F)) + F**(a*c + b*
c*x)*f**2*cos(I*b*c*x*log(F)/2 - d)**2/(b*c*log(F)) + 8*F**(a*c + b*c*x)*f**2*cos(I*b*c*x*log(F)/2 - d)/(3*b*c
*log(F)) + F**(a*c + b*c*x)*f**2/(b*c*log(F)), Eq(e, -I*b*c*log(F)/2)), (-F**(a*c + b*c*x)*f**2*x*sin(I*b*c*x*
log(F)/2 + d)**2/4 + I*F**(a*c + b*c*x)*f**2*x*sin(I*b*c*x*log(F)/2 + d)*cos(I*b*c*x*log(F)/2 + d)/2 + F**(a*c
+ b*c*x)*f**2*x*cos(I*b*c*x*log(F)/2 + d)**2/4 + F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F)/2 + d)**2/(b*c*log(
F)) - 3*I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F)/2 + d)*cos(I*b*c*x*log(F)/2 + d)/(2*b*c*log(F)) + 4*I*F**(a
*c + b*c*x)*f**2*sin(I*b*c*x*log(F)/2 + d)/(3*b*c*log(F)) + 8*F**(a*c + b*c*x)*f**2*cos(I*b*c*x*log(F)/2 + d)/
(3*b*c*log(F)) + F**(a*c + b*c*x)*f**2/(b*c*log(F)), Eq(e, I*b*c*log(F)/2)), (I*F**(a*c + b*c*x)*f**2*x*sin(I*
b*c*x*log(F) + d) + F**(a*c + b*c*x)*f**2*x*cos(I*b*c*x*log(F) + d) + 2*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(
F) + d)**2/(3*b*c*log(F)) - 2*I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F) + d)*cos(I*b*c*x*log(F) + d)/(3*b*c*l
og(F)) - I*F**(a*c + b*c*x)*f**2*sin(I*b*c*x*log(F) + d)/(b*c*log(F)) + F**(a*c + b*c*x)*f**2*cos(I*b*c*x*log(
F) + d)**2/(3*b*c*log(F)) + F**(a*c + b*c*x)*f**2/(b*c*log(F)), Eq(e, I*b*c*log(F))), (F**(a*c + b*c*x)*b**4*c
**4*f**2*log(F)**4*cos(d + e*x)**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*
F**(a*c + b*c*x)*b**4*c**4*f**2*log(F)**4*cos(d + e*x)/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b
*c*e**4*log(F)) + F**(a*c + b*c*x)*b**4*c**4*f**2*log(F)**4/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3
+ 4*b*c*e**4*log(F)) + 2*F**(a*c + b*c*x)*b**3*c**3*e*f**2*log(F)**3*sin(d + e*x)*cos(d + e*x)/(b**5*c**5*log(
F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c + b*c*x)*b**3*c**3*e*f**2*log(F)**3*sin(d
+ e*x)/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c + b*c*x)*b**2*c**2*e
**2*f**2*log(F)**2*sin(d + e*x)**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 3*
F**(a*c + b*c*x)*b**2*c**2*e**2*f**2*log(F)**2*cos(d + e*x)**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)*
*3 + 4*b*c*e**4*log(F)) + 8*F**(a*c + b*c*x)*b**2*c**2*e**2*f**2*log(F)**2*cos(d + e*x)/(b**5*c**5*log(F)**5 +
5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 5*F**(a*c + b*c*x)*b**2*c**2*e**2*f**2*log(F)**2/(b**5*c**5
*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c + b*c*x)*b*c*e**3*f**2*log(F)*sin(d +
e*x)*cos(d + e*x)/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 8*F**(a*c + b*c*x)
*b*c*e**3*f**2*log(F)*sin(d + e*x)/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*
F**(a*c + b*c*x)*e**4*f**2*sin(d + e*x)**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(
F)) + 2*F**(a*c + b*c*x)*e**4*f**2*cos(d + e*x)**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e
**4*log(F)) + 4*F**(a*c + b*c*x)*e**4*f**2/(b**5*c**5*log(F)**5 + 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(
F)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (245) = 490\).

Time = 0.24 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.36 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {{\left ({\left (F^{a c} b^{2} c^{2} \cos \left (2 \, d\right ) \log \left (F\right )^{2} + 2 \, F^{a c} b c e \log \left (F\right ) \sin \left (2 \, d\right )\right )} F^{b c x} \cos \left (2 \, e x\right ) + {\left (F^{a c} b^{2} c^{2} \cos \left (2 \, d\right ) \log \left (F\right )^{2} - 2 \, F^{a c} b c e \log \left (F\right ) \sin \left (2 \, d\right )\right )} F^{b c x} \cos \left (2 \, e x + 4 \, d\right ) - {\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} \sin \left (2 \, d\right ) - 2 \, F^{a c} b c e \cos \left (2 \, d\right ) \log \left (F\right )\right )} F^{b c x} \sin \left (2 \, e x\right ) + {\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} \sin \left (2 \, d\right ) + 2 \, F^{a c} b c e \cos \left (2 \, d\right ) \log \left (F\right )\right )} F^{b c x} \sin \left (2 \, e x + 4 \, d\right ) + 2 \, {\left (F^{a c} b^{2} c^{2} \cos \left (2 \, d\right )^{2} \log \left (F\right )^{2} + F^{a c} b^{2} c^{2} \log \left (F\right )^{2} \sin \left (2 \, d\right )^{2} + 4 \, {\left (F^{a c} \cos \left (2 \, d\right )^{2} + F^{a c} \sin \left (2 \, d\right )^{2}\right )} e^{2}\right )} F^{b c x}\right )} f^{2}}{4 \, {\left (b^{3} c^{3} \cos \left (2 \, d\right )^{2} \log \left (F\right )^{3} + b^{3} c^{3} \log \left (F\right )^{3} \sin \left (2 \, d\right )^{2} + 4 \, {\left (b c \cos \left (2 \, d\right )^{2} \log \left (F\right ) + b c \log \left (F\right ) \sin \left (2 \, d\right )^{2}\right )} e^{2}\right )}} + \frac {{\left ({\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) - F^{a c} e \sin \left (d\right )\right )} F^{b c x} \cos \left (e x + 2 \, d\right ) + {\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) + F^{a c} e \sin \left (d\right )\right )} F^{b c x} \cos \left (e x\right ) + {\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) + F^{a c} e \cos \left (d\right )\right )} F^{b c x} \sin \left (e x + 2 \, d\right ) - {\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) - F^{a c} e \cos \left (d\right )\right )} F^{b c x} \sin \left (e x\right )\right )} f^{2}}{b^{2} c^{2} \cos \left (d\right )^{2} \log \left (F\right )^{2} + b^{2} c^{2} \log \left (F\right )^{2} \sin \left (d\right )^{2} + {\left (\cos \left (d\right )^{2} + \sin \left (d\right )^{2}\right )} e^{2}} + \frac {F^{b c x + a c} f^{2}}{b c \log \left (F\right )} \]

[In]

integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="maxima")

[Out]

1/4*((F^(a*c)*b^2*c^2*cos(2*d)*log(F)^2 + 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*e*x) + (F^(a*c)*b^2
*c^2*cos(2*d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*e*x + 4*d) - (F^(a*c)*b^2*c^2*log(F)
^2*sin(2*d) - 2*F^(a*c)*b*c*e*cos(2*d)*log(F))*F^(b*c*x)*sin(2*e*x) + (F^(a*c)*b^2*c^2*log(F)^2*sin(2*d) + 2*F
^(a*c)*b*c*e*cos(2*d)*log(F))*F^(b*c*x)*sin(2*e*x + 4*d) + 2*(F^(a*c)*b^2*c^2*cos(2*d)^2*log(F)^2 + F^(a*c)*b^
2*c^2*log(F)^2*sin(2*d)^2 + 4*(F^(a*c)*cos(2*d)^2 + F^(a*c)*sin(2*d)^2)*e^2)*F^(b*c*x))*f^2/(b^3*c^3*cos(2*d)^
2*log(F)^3 + b^3*c^3*log(F)^3*sin(2*d)^2 + 4*(b*c*cos(2*d)^2*log(F) + b*c*log(F)*sin(2*d)^2)*e^2) + ((F^(a*c)*
b*c*cos(d)*log(F) - F^(a*c)*e*sin(d))*F^(b*c*x)*cos(e*x + 2*d) + (F^(a*c)*b*c*cos(d)*log(F) + F^(a*c)*e*sin(d)
)*F^(b*c*x)*cos(e*x) + (F^(a*c)*b*c*log(F)*sin(d) + F^(a*c)*e*cos(d))*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b*c*
log(F)*sin(d) - F^(a*c)*e*cos(d))*F^(b*c*x)*sin(e*x))*f^2/(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*log(F)^2*sin(d)
^2 + (cos(d)^2 + sin(d)^2)*e^2) + F^(b*c*x + a*c)*f^2/(b*c*log(F))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 1736, normalized size of antiderivative = 7.09 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\text {Too large to display} \]

[In]

integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="giac")

[Out]

1/2*(2*b*c*f^2*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)*log(abs(
F))/(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)*f^2*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))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 2*(2*b*c*f^2*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 + e*x + d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(
F) - pi*b*c + 2*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 2*e)*f^2*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 + e*x + d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 2*e)^2))*e^(b*c*x*log(abs
(F)) + a*c*log(abs(F))) + 2*(2*b*c*f^2*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
- e*x - d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 2*e)^2) + (pi*b*c*sgn(F) - pi*b*c
- 2*e)*f^2*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 - e*x - d)/(4*b^2*c^2*log(
abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 2*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 1/2*(2*b*c*f^2*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)*log(abs(F))/(4*b^2*c^2*log(ab
s(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 4*e)^2) + (pi*b*c*sgn(F) - pi*b*c - 4*e)*f^2*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))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 3*(2*b*c*f^2*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)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(
F) - pi*b*c)*f^2*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)/(4*b^2*c^2*log(abs(
F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*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 + 2*I*e*x + 2*I*d)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c +
8*b*c*log(abs(F)) + 16*I*e) - I*f^2*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*p
i*a*c - 2*I*e*x - 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*I*e))*e^(b*c*x*log(abs(F))
+ a*c*log(abs(F))) + I*(I*f^2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*e) - I*f^2*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)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F
)) - 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) - 2*I*e) - I*f
^2*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)/(-I*pi*b*c*s
gn(F) + I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*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 - 2*I*e*x - 2*I*d)/(4*I*pi*b*c*sgn(F) - 4*I
*pi*b*c + 8*b*c*log(abs(F)) - 16*I*e) - I*f^2*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 + 2*I*e*x + 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*I*e))*e^(b*c*x*lo
g(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*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)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) - I*f^2*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)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F))))*e^(b*c*x*log(
abs(F)) + a*c*log(abs(F))) + I*(I*f^2*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*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))) - I*f^2*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*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) +
a*c*log(abs(F)))

Mupad [B] (verification not implemented)

Time = 27.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx=\frac {F^{a\,c+b\,c\,x}\,f^2\,\left (6\,e^4+\frac {3\,b^4\,c^4\,{\ln \left (F\right )}^4}{2}+2\,b^4\,c^4\,\cos \left (d+e\,x\right )\,{\ln \left (F\right )}^4+\frac {b^4\,c^4\,{\ln \left (F\right )}^4\,\cos \left (2\,d+2\,e\,x\right )}{2}+\frac {15\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2}{2}+8\,b\,c\,e^3\,\sin \left (d+e\,x\right )\,\ln \left (F\right )+8\,b^2\,c^2\,e^2\,\cos \left (d+e\,x\right )\,{\ln \left (F\right )}^2+b^3\,c^3\,e\,{\ln \left (F\right )}^3\,\sin \left (2\,d+2\,e\,x\right )+b\,c\,e^3\,\ln \left (F\right )\,\sin \left (2\,d+2\,e\,x\right )+\frac {b^2\,c^2\,e^2\,{\ln \left (F\right )}^2\,\cos \left (2\,d+2\,e\,x\right )}{2}+2\,b^3\,c^3\,e\,\sin \left (d+e\,x\right )\,{\ln \left (F\right )}^3\right )}{b\,c\,\ln \left (F\right )\,\left (b^4\,c^4\,{\ln \left (F\right )}^4+5\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+4\,e^4\right )} \]

[In]

int(F^(c*(a + b*x))*(f + f*cos(d + e*x))^2,x)

[Out]

(F^(a*c + b*c*x)*f^2*(6*e^4 + (3*b^4*c^4*log(F)^4)/2 + 2*b^4*c^4*cos(d + e*x)*log(F)^4 + (b^4*c^4*log(F)^4*cos
(2*d + 2*e*x))/2 + (15*b^2*c^2*e^2*log(F)^2)/2 + 8*b*c*e^3*sin(d + e*x)*log(F) + 8*b^2*c^2*e^2*cos(d + e*x)*lo
g(F)^2 + b^3*c^3*e*log(F)^3*sin(2*d + 2*e*x) + b*c*e^3*log(F)*sin(2*d + 2*e*x) + (b^2*c^2*e^2*log(F)^2*cos(2*d
+ 2*e*x))/2 + 2*b^3*c^3*e*sin(d + e*x)*log(F)^3))/(b*c*log(F)*(4*e^4 + b^4*c^4*log(F)^4 + 5*b^2*c^2*e^2*log(F
)^2))