\(\int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx\) [32]

Optimal result

Integrand size = 42, antiderivative size = 23 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=f F^{c (a+b x)} x (f x)^m \sin (d+e x) \]

[Out]

f*F^(c*(b*x+a))*x*(f*x)^m*sin(e*x+d)

Rubi [F]

\[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=\int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx \]

[In]

Int[f*F^(c*(a + b*x))*(f*x)^m*(e*x*Cos[d + e*x] + (1 + m + b*c*x*Log[F])*Sin[d + e*x]),x]

[Out]

e*Defer[Int][F^(a*c + b*c*x)*(f*x)^(1 + m)*Cos[d + e*x], x] + f*(1 + m)*Defer[Int][F^(a*c + b*c*x)*(f*x)^m*Sin
[d + e*x], x] + b*c*Log[F]*Defer[Int][F^(a*c + b*c*x)*(f*x)^(1 + m)*Sin[d + e*x], x]

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=f F^{a c+b c x} x (f x)^m \sin (d+e x) \]

[In]

Integrate[f*F^(c*(a + b*x))*(f*x)^m*(e*x*Cos[d + e*x] + (1 + m + b*c*x*Log[F])*Sin[d + e*x]),x]

[Out]

f*F^(a*c + b*c*x)*x*(f*x)^m*Sin[d + e*x]

Maple [A] (verified)

Time = 5.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(f \,F^{c \left (x b +a \right )} x \left (f x \right )^{m} \sin \left (e x +d \right )\)	\(24\)
risch	\(-\frac {i x^{m} f^{m} F^{c \left (x b +a \right )} x f \left ({\mathrm e}^{i e x} {\mathrm e}^{i d} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i f x \right )^{3} m}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i f \right ) m}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i x \right ) m}{2}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i f x \right ) \operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x \right ) m}{2}}-{\mathrm e}^{-i e x} {\mathrm e}^{-i d} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i f x \right )^{3} m}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i f \right ) m}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i f x \right )^{2} \operatorname {csgn}\left (i x \right ) m}{2}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i f x \right ) \operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x \right ) m}{2}}\right )}{2}\)	\(195\)

[In]

int(f*F^(c*(b*x+a))*(f*x)^m*(e*x*cos(e*x+d)+(1+m+b*c*x*ln(F))*sin(e*x+d)),x,method=_RETURNVERBOSE)

[Out]

f*F^(c*(b*x+a))*x*(f*x)^m*sin(e*x+d)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=\left (f x\right )^{m} F^{b c x + a c} f x \sin \left (e x + d\right ) \]

[In]

integrate(f*F^(c*(b*x+a))*(f*x)^m*(e*x*cos(e*x+d)+(1+m+b*c*x*log(F))*sin(e*x+d)),x, algorithm="fricas")

[Out]

(f*x)^m*F^(b*c*x + a*c)*f*x*sin(e*x + d)

Sympy [F]

\[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=f \left (\int F^{a c + b c x} \left (f x\right )^{m} \sin {\left (d + e x \right )}\, dx + \int F^{a c + b c x} m \left (f x\right )^{m} \sin {\left (d + e x \right )}\, dx + \int F^{a c + b c x} e x \left (f x\right )^{m} \cos {\left (d + e x \right )}\, dx + \int F^{a c + b c x} b c x \left (f x\right )^{m} \log {\left (F \right )} \sin {\left (d + e x \right )}\, dx\right ) \]

[In]

integrate(f*F**(c*(b*x+a))*(f*x)**m*(e*x*cos(e*x+d)+(1+m+b*c*x*ln(F))*sin(e*x+d)),x)

[Out]

f*(Integral(F**(a*c + b*c*x)*(f*x)**m*sin(d + e*x), x) + Integral(F**(a*c + b*c*x)*m*(f*x)**m*sin(d + e*x), x)
+ Integral(F**(a*c + b*c*x)*e*x*(f*x)**m*cos(d + e*x), x) + Integral(F**(a*c + b*c*x)*b*c*x*(f*x)**m*log(F)*s
in(d + e*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=F^{a c} f^{m + 1} x e^{\left (b c x \log \left (F\right ) + m \log \left (x\right )\right )} \sin \left (e x + d\right ) \]

[In]

integrate(f*F^(c*(b*x+a))*(f*x)^m*(e*x*cos(e*x+d)+(1+m+b*c*x*log(F))*sin(e*x+d)),x, algorithm="maxima")

[Out]

F^(a*c)*f^(m + 1)*x*e^(b*c*x*log(F) + m*log(x))*sin(e*x + d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4746 vs. \(2 (23) = 46\).

Time = 0.53 (sec) , antiderivative size = 4746, normalized size of antiderivative = 206.35 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=\text {Too large to display} \]

[In]

integrate(f*F^(c*(b*x+a))*(f*x)^m*(e*x*cos(e*x+d)+(1+m+b*c*x*log(F))*sin(e*x+d)),x, algorithm="giac")

[Out]

(x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*flo
or(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*
sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*
m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) - x*a
bs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-
1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(
F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m -
1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + x*abs(F
)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*
sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) -
1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*
e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 - x*abs(F)^(a
*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(
f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)*tan(1/4*pi*b*c*x*sgn(F) - 1/4*p
i*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2
*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 - x*abs(F)^(a*c)*
e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) -
1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*
b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*t
an(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d) - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1
/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(
x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) +
1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2 - x*abs
(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/
4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)*tan(1/4*pi*b*c*x*sgn(F) -
1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*
e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2 + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(
x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4
*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn
(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*
d) - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m
*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*
c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) + x*abs(F)^(a*c)*e^(b*c*x*log(abs
(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1)
+ 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan
(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4
*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x)
- 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1
/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + x*abs(F
)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*
sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)*tan(1/4*pi*b*c*x*sgn(F) - 1
/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*
x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)
))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*p
i*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*
a*c - 1/2*d)^2 + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*
b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*t
an(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 - x*abs(F)^(a*c)*e^(b
*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4
*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/
2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(
x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4
*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*p
i*a*c - 1/2*d)^2 - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*p
i*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2
*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*
m*sgn(x) - 1/2*pi*m - 1/2*e*x) + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*
sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*
m + 1/2*e*x)*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(
f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2 - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan
(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sg
n(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d) + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F))
+ m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1
/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d) - x*abs(F
)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*
sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4
*pi*a*c + 1/2*d)^2 - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4
*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)
*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2 - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*t
an(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*
sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F
)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) +
1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) - x*abs
(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*p
i*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d) + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*
c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2
*pi*m + 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(a
bs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sg
n(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + x*abs(F)^(a*c)*e^
(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)*tan(1/4*pi*a*c*sgn(F) -
1/4*pi*a*c - 1/2*d)^2 + x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) -
1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*
e*x) - x*abs(F)^(a*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi
*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x) + x*abs(F)^(a
*c)*e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d) - x*abs(F)^(a*c)*
e^(b*c*x*log(abs(F)) + m*log(abs(f)*abs(x)))*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d))*f/(tan(1/4*pi*b*c*x*
sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*
m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sg
n(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sg
n(F) - 1/4*pi*a*c - 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) +
1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*f
loor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*
sgn(F) - 1/4*pi*a*c + 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x)
+ 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m
*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*
c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x
) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)
^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sg
n(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4
*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi
*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*
b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1
/2*pi*m - 1/2*e*x)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4
*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2 + tan(1/4
*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x)
- 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*
x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m + 1/2*e*x)^2*tan(1
/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1
/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c
- 1/2*d)^2 + tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2*tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + tan
(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sg
n(x) - 1/2*pi*m + 1/2*e*x)^2 + tan(1/4*pi*b*c*x*sgn(F) - 1/4*pi*b*c*x + pi*m*floor(-1/4*sgn(f) - 1/4*sgn(x) +
1) + 1/4*pi*m*sgn(f) + 1/4*pi*m*sgn(x) - 1/2*pi*m - 1/2*e*x)^2 + tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c + 1/2*d)^2
+ tan(1/4*pi*a*c*sgn(F) - 1/4*pi*a*c - 1/2*d)^2 + 1)

Mupad [B] (verification not implemented)

Time = 28.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int f F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(1+m+b c x \log (F)) \sin (d+e x)) \, dx=F^{c\,\left (a+b\,x\right )}\,f\,x\,\sin \left (d+e\,x\right )\,{\left (f\,x\right )}^m \]

[In]

int(F^(c*(a + b*x))*f*(f*x)^m*(sin(d + e*x)*(m + b*c*x*log(F) + 1) + e*x*cos(d + e*x)),x)

[Out]

F^(c*(a + b*x))*f*x*sin(d + e*x)*(f*x)^m