\(\int F^{c (a+b x)} \cos ^3(d+e x) \, dx\) [11]

Optimal result

Integrand size = 18, antiderivative size = 199 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 F^{c (a+b x)} \cos (d+e x) \log (F)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sin (d+e x)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)} \]

[Out]

b*c*F^(c*(b*x+a))*cos(e*x+d)^3*ln(F)/(9*e^2+b^2*c^2*ln(F)^2)+6*b*c*e^2*F^(c*(b*x+a))*cos(e*x+d)*ln(F)/(9*e^4+1
0*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)+3*e*F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d)/(9*e^2+b^2*c^2*ln(F)^2)+6*e^3
*F^(c*(b*x+a))*sin(e*x+d)/(9*e^4+10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4520, 4518} \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {b c \log (F) \cos ^3(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {3 e \sin (d+e x) \cos ^2(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {6 b c e^2 \log (F) \cos (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)+10 b^2 c^2 e^2 \log ^2(F)+9 e^4}+\frac {6 e^3 \sin (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)+10 b^2 c^2 e^2 \log ^2(F)+9 e^4} \]

[In]

Int[F^(c*(a + b*x))*Cos[d + e*x]^3,x]

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {\left (6 e^2\right ) \int F^{c (a+b x)} \cos (d+e x) \, dx}{9 e^2+b^2 c^2 \log ^2(F)} \\ & = \frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 F^{c (a+b x)} \cos (d+e x) \log (F)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sin (d+e x)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (b c \cos (3 (d+e x)) \log (F) \left (e^2+b^2 c^2 \log ^2(F)\right )+3 b c \cos (d+e x) \log (F) \left (9 e^2+b^2 c^2 \log ^2(F)\right )+6 e \left (5 e^2+b^2 c^2 \log ^2(F)+\cos (2 (d+e x)) \left (e^2+b^2 c^2 \log ^2(F)\right )\right ) \sin (d+e x)\right )}{4 \left (9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \]

[In]

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

[Out]

(F^(c*(a + b*x))*(b*c*Cos[3*(d + e*x)]*Log[F]*(e^2 + b^2*c^2*Log[F]^2) + 3*b*c*Cos[d + e*x]*Log[F]*(9*e^2 + b^
2*c^2*Log[F]^2) + 6*e*(5*e^2 + b^2*c^2*Log[F]^2 + Cos[2*(d + e*x)]*(e^2 + b^2*c^2*Log[F]^2))*Sin[d + e*x]))/(4
*(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {F^{c \left (x b +a \right )} \left (b c \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \cos \left (3 e x +3 d \right )+\left (3 \ln \left (F \right )^{2} b^{2} c^{2} e +3 e^{3}\right ) \sin \left (3 e x +3 d \right )+3 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (\cos \left (e x +d \right ) \ln \left (F \right ) b c +e \sin \left (e x +d \right )\right )\right )}{4 b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+36 e^{4}}\)	\(144\)
risch	\(\frac {3 b c \,F^{c \left (x b +a \right )} \cos \left (e x +d \right ) \ln \left (F \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {3 e \,F^{c \left (x b +a \right )} \sin \left (e x +d \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {F^{c \left (x b +a \right )} b c \ln \left (F \right ) \cos \left (3 e x +3 d \right )}{4 b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}}+\frac {3 e \,F^{c \left (x b +a \right )} \sin \left (3 e x +3 d \right )}{4 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}\)	\(158\)
norman	\(\frac {\frac {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \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 {12 e \left (b^{2} c^{2} \ln \left (F \right )^{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 )^{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 {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 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 )}{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 {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 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 )^{5}}{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 {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 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 )^{6}}{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 {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{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}}{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 {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{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 )^{4}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\)	\(541\)

[In]

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

[Out]

F^(c*(b*x+a))*(b*c*ln(F)*(e^2+b^2*c^2*ln(F)^2)*cos(3*e*x+3*d)+(3*ln(F)^2*b^2*c^2*e+3*e^3)*sin(3*e*x+3*d)+3*(9*
e^2+b^2*c^2*ln(F)^2)*(cos(e*x+d)*ln(F)*b*c+e*sin(e*x+d)))/(4*b^4*c^4*ln(F)^4+40*b^2*c^2*e^2*ln(F)^2+36*e^4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {{\left (b^{3} c^{3} \cos \left (e x + d\right )^{3} \log \left (F\right )^{3} + {\left (b c e^{2} \cos \left (e x + d\right )^{3} + 6 \, b c e^{2} \cos \left (e x + d\right )\right )} \log \left (F\right ) + 3 \, {\left (b^{2} c^{2} e \cos \left (e x + d\right )^{2} \log \left (F\right )^{2} + e^{3} \cos \left (e x + d\right )^{2} + 2 \, e^{3}\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 10 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 9 \, e^{4}} \]

[In]

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

[Out]

(b^3*c^3*cos(e*x + d)^3*log(F)^3 + (b*c*e^2*cos(e*x + d)^3 + 6*b*c*e^2*cos(e*x + d))*log(F) + 3*(b^2*c^2*e*cos
(e*x + d)^2*log(F)^2 + e^3*cos(e*x + d)^2 + 2*e^3)*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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.23 (sec) , antiderivative size = 1671, normalized size of antiderivative = 8.40 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((x*cos(d)**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*cos(d)**3, Eq(b, 0) & Eq(e, 0)), (x*cos(d)**3, Eq(c,
0) & Eq(e, 0)), (3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*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)*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)*x*cos(I*b*c*x*log(F) - d)**3/8 - 5*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)**3/(8*b
*c*log(F)) - F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/(4*b*c*log(F)) - I*F**(a*c +
b*c*x)*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)**2/(b*c*log(F)) - 3*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)
- d)**3/(8*b*c*log(F)), Eq(e, -I*b*c*log(F))), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 - d)**3/8 - 3*F**(
a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 - d)**2*cos(I*b*c*x*log(F)/3 - d)/8 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*l
og(F)/3 - d)*cos(I*b*c*x*log(F)/3 - d)**2/8 + F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/3 - d)**3/8 + I*F**(a*c +
b*c*x)*sin(I*b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)) + 3*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 - d)*cos(I*b*c*
x*log(F)/3 - d)**2/(4*b*c*log(F)) + 9*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)), Eq(e, -I*b
*c*log(F)/3)), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 + d)**3/8 - 3*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)
/3 + d)**2*cos(I*b*c*x*log(F)/3 + d)/8 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 + d)*cos(I*b*c*x*log(F)/3
+ d)**2/8 + F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/3 + d)**3/8 + I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 + d)*
*3/(8*b*c*log(F)) + 3*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 + d)*cos(I*b*c*x*log(F)/3 + d)**2/(4*b*c*log(F))
+ 9*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/3 + d)**3/(8*b*c*log(F)), Eq(e, I*b*c*log(F)/3)), (3*I*F**(a*c + b*c*
x)*x*sin(I*b*c*x*log(F) + d)**3/8 + 3*F**(a*c + b*c*x)*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)*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)*x*cos(I*b*c
*x*log(F) + d)**3/8 - 5*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F) + d)**3/(8*b*c*log(F)) - F**(a*c + b*c*x)*sin(I*
b*c*x*log(F) + d)**2*cos(I*b*c*x*log(F) + d)/(4*b*c*log(F)) - I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F) + d)*cos(I
*b*c*x*log(F) + d)**2/(b*c*log(F)) - 3*F**(a*c + b*c*x)*cos(I*b*c*x*log(F) + d)**3/(8*b*c*log(F)), Eq(e, I*b*c
*log(F))), (F**(a*c + b*c*x)*b**3*c**3*log(F)**3*cos(d + e*x)**3/(b**4*c**4*log(F)**4 + 10*b**2*c**2*e**2*log(
F)**2 + 9*e**4) + 3*F**(a*c + b*c*x)*b**2*c**2*e*log(F)**2*sin(d + e*x)*cos(d + e*x)**2/(b**4*c**4*log(F)**4 +
10*b**2*c**2*e**2*log(F)**2 + 9*e**4) + 6*F**(a*c + b*c*x)*b*c*e**2*log(F)*sin(d + e*x)**2*cos(d + e*x)/(b**4
*c**4*log(F)**4 + 10*b**2*c**2*e**2*log(F)**2 + 9*e**4) + 7*F**(a*c + b*c*x)*b*c*e**2*log(F)*cos(d + e*x)**3/(
b**4*c**4*log(F)**4 + 10*b**2*c**2*e**2*log(F)**2 + 9*e**4) + 6*F**(a*c + b*c*x)*e**3*sin(d + e*x)**3/(b**4*c*
*4*log(F)**4 + 10*b**2*c**2*e**2*log(F)**2 + 9*e**4) + 9*F**(a*c + b*c*x)*e**3*sin(d + e*x)*cos(d + e*x)**2/(b
**4*c**4*log(F)**4 + 10*b**2*c**2*e**2*log(F)**2 + 9*e**4), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (199) = 398\).

Time = 0.25 (sec) , antiderivative size = 813, normalized size of antiderivative = 4.09 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {{\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} + 3 \, F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) + 3 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (3 \, e x\right ) + {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) - 3 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (3 \, e x + 6 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} - F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + 9 \, F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) - 9 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (e x + 4 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} + F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + 9 \, F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) + 9 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (e x - 2 \, d\right ) - {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) - 3 \, F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) - 3 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (3 \, e x\right ) + {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) + 3 \, F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) + 3 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (3 \, e x + 6 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) + F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + 9 \, F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) + 9 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (e x + 4 \, d\right ) - 3 \, {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) - F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + 9 \, F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) - 9 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (e x - 2 \, d\right )}{8 \, {\left (b^{4} c^{4} \cos \left (3 \, d\right )^{2} \log \left (F\right )^{4} + b^{4} c^{4} \log \left (F\right )^{4} \sin \left (3 \, d\right )^{2} + 9 \, {\left (\cos \left (3 \, d\right )^{2} + \sin \left (3 \, d\right )^{2}\right )} e^{4} + 10 \, {\left (b^{2} c^{2} \cos \left (3 \, d\right )^{2} \log \left (F\right )^{2} + b^{2} c^{2} \log \left (F\right )^{2} \sin \left (3 \, d\right )^{2}\right )} e^{2}\right )}} \]

[In]

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

[Out]

1/8*((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)*cos(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)*cos(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*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)*cos(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*lo
g(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)*cos(e*x - 2*d) - (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)*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)*sin(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*cos(3*d)*l
og(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)*sin(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)*sin(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)*sin(e*x - 2*d))/(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)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.39 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*b*c*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 + 3*e*x + 3*d)*log(abs(F))/
(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 6*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 6*e)*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 + 3*e*x + 3*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(
F) - pi*b*c + 6*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 3/4*(2*b*c*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)*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))) + 3/4*(2*b*c*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)*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))) + 1/4*(2*b*c*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 - 3*e*x - 3*d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F)
- pi*b*c - 6*e)^2) + (pi*b*c*sgn(F) - pi*b*c - 6*e)*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 - 3*e*x - 3*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 6*e)^2))*e^(b*c*x*log(abs(F
)) + a*c*log(abs(F))) + I*(I*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) - I*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))) + 3*I*(I*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) - I*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*(I*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) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*p
i*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))) + I*(I*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) - I*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)))

Mupad [B] (verification not implemented)

Time = 28.71 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.96 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )+\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,\left (e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )-\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,3{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )+\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (3\,e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {3\,F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )-\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,1{}\mathrm {i}\right )} \]

[In]

int(F^(c*(a + b*x))*cos(d + e*x)^3,x)

[Out]

- (F^(c*(a + b*x))*(cos(e*x) + sin(e*x)*1i)*(cos(d) + sin(d)*1i)*3i)/(8*(e - b*c*log(F)*1i)) - (F^(c*(a + b*x)
)*(cos(3*e*x) - sin(3*e*x)*1i)*(cos(3*d) - sin(3*d)*1i))/(8*(e*3i - b*c*log(F))) - (F^(c*(a + b*x))*(cos(3*e*x
) + sin(3*e*x)*1i)*(cos(3*d) + sin(3*d)*1i)*1i)/(8*(3*e - b*c*log(F)*1i)) - (3*F^(c*(a + b*x))*(cos(e*x) - sin
(e*x)*1i)*(cos(d) - sin(d)*1i))/(8*(e*1i - b*c*log(F)))