\(\int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx\) [34]

Optimal result

Integrand size = 35, antiderivative size = 17 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=F^{c (a+b x)} x \sin (d+e x) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(17)=34\).

Time = 1.12 (sec) , antiderivative size = 327, normalized size of antiderivative = 19.24, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6873, 6874, 4518, 4554, 4517, 4553} \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=\frac {b^2 c^2 x \log ^2(F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {e^2 x \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {b c \log (F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac {b c e^2 \log (F) \sin (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}-\frac {e \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {b^2 c^2 e \log ^2(F) \cos (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}+\frac {e^3 \cos (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2}-\frac {b^3 c^3 \log ^3(F) \sin (d+e x) F^{a c+b c x}}{\left (b^2 c^2 \log ^2(F)+e^2\right )^2} \]

[In]

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

[Out]

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

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[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 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 4553

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.)*Sin[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 4554

Int[Cos[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

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^{a c+b c x} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx \\ & = \int \left (e F^{a c+b c x} x \cos (d+e x)+F^{a c+b c x} (1+b c x \log (F)) \sin (d+e x)\right ) \, dx \\ & = e \int F^{a c+b c x} x \cos (d+e x) \, dx+\int F^{a c+b c x} (1+b c x \log (F)) \sin (d+e x) \, dx \\ & = \frac {b c e F^{a c+b c x} x \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-e \int \left (\frac {b c F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {e F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\right ) \, dx+\int \left (F^{a c+b c x} \sin (d+e x)+b c F^{a c+b c x} x \log (F) \sin (d+e x)\right ) \, dx \\ & = \frac {b c e F^{a c+b c x} x \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+(b c \log (F)) \int F^{a c+b c x} x \sin (d+e x) \, dx-\frac {e^2 \int F^{a c+b c x} \sin (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}-\frac {(b c e \log (F)) \int F^{a c+b c x} \cos (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}+\int F^{a c+b c x} \sin (d+e x) \, dx \\ & = \frac {e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac {2 b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac {e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-(b c \log (F)) \int \left (-\frac {e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\right ) \, dx \\ & = \frac {e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac {2 b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac {e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {(b c e \log (F)) \int F^{a c+b c x} \cos (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)}-\frac {\left (b^2 c^2 \log ^2(F)\right ) \int F^{a c+b c x} \sin (d+e x) \, dx}{e^2+b^2 c^2 \log ^2(F)} \\ & = \frac {e^3 F^{a c+b c x} \cos (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac {b^2 c^2 e F^{a c+b c x} \cos (d+e x) \log ^2(F)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {e F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac {b c e^2 F^{a c+b c x} \log (F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}-\frac {b^3 c^3 F^{a c+b c x} \log ^3(F) \sin (d+e x)}{\left (e^2+b^2 c^2 \log ^2(F)\right )^2}+\frac {e^2 F^{a c+b c x} x \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b^2 c^2 F^{a c+b c x} x \log ^2(F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=F^{a c+b c x} x \sin (d+e x) \]

[In]

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

[Out]

F^(a*c + b*c*x)*x*Sin[d + e*x]

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
risch	\(F^{c \left (x b +a \right )} x \sin \left (e x +d \right )\)	\(18\)
parallelrisch	\(F^{c \left (x b +a \right )} x \sin \left (e x +d \right )\)	\(18\)
norman	\(\frac {2 x \,{\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}\)	\(38\)
parts	\(\frac {\frac {e \,{\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}}-\frac {e \,{\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 b c \ln \left (F \right ) {\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}}}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}+\frac {b c \ln \left (F \right ) \left (\frac {e x \,{\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}}-\frac {e x \,{\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 \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 )}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}+\frac {2 e b c \ln \left (F \right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}+\frac {2 b c \ln \left (F \right ) x \,{\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 e b c \ln \left (F \right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}\right )}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}+\frac {\frac {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 )^{2}}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}+\frac {e b c \ln \left (F \right ) x \,{\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {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 )}}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}+\frac {2 e^{2} x \,{\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 {4 e^{2} b c \ln \left (F \right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )^{2}}-\frac {e b c \ln \left (F \right ) x \,{\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}}\)	\(689\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=F^{b c x + a c} x \sin \left (e x + d\right ) \]

[In]

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

[Out]

F^(b*c*x + a*c)*x*sin(e*x + d)

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=F^{a c + b c x} x \sin {\left (d + e x \right )} \]

[In]

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

[Out]

F**(a*c + b*c*x)*x*sin(d + e*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1382 vs. \(2 (17) = 34\).

Time = 0.33 (sec) , antiderivative size = 1382, normalized size of antiderivative = 81.29 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((F^(a*c)*b^2*c^2*log(F)^2*sin(d) + 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*
log(F)^3*sin(d) + F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) + F^(a*c)*e^3*cos(d))*x)*F
^(b*c*x)*cos(e*x + 2*d) - (F^(a*c)*b^2*c^2*log(F)^2*sin(d) - 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d
) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) - F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) - F^(
a*c)*e^3*cos(d))*x)*F^(b*c*x)*cos(e*x) - (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(d) - F^
(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 - F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*cos(
d)*log(F) - F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 + 2*F^(a*c)*b*c
*e*log(F)*sin(d) - F^(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 + F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) +
F^(a*c)*b*c*e^2*cos(d)*log(F) + F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*sin(e*x))*b*c*log(F)/(b^4*c^4*cos(d)^2*log(F
)^4 + b^4*c^4*log(F)^4*sin(d)^2 + (cos(d)^2 + sin(d)^2)*e^4 + 2*(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*log(F)^2*
sin(d)^2)*e^2) - 1/2*((F^(a*c)*b^2*c^2*cos(d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(d) - F^(a*c)*e^2*cos(d) -
(F^(a*c)*b^3*c^3*cos(d)*log(F)^3 - F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*cos(d)*log(F) - F^(a*c)
*e^3*sin(d))*x)*F^(b*c*x)*cos(e*x + 2*d) + (F^(a*c)*b^2*c^2*cos(d)*log(F)^2 + 2*F^(a*c)*b*c*e*log(F)*sin(d) -
F^(a*c)*e^2*cos(d) - (F^(a*c)*b^3*c^3*cos(d)*log(F)^3 + F^(a*c)*b^2*c^2*e*log(F)^2*sin(d) + F^(a*c)*b*c*e^2*co
s(d)*log(F) + F^(a*c)*e^3*sin(d))*x)*F^(b*c*x)*cos(e*x) + (F^(a*c)*b^2*c^2*log(F)^2*sin(d) + 2*F^(a*c)*b*c*e*c
os(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) + F^(a*c)*b^2*c^2*e*cos(d)*log(F)^2 + F^(
a*c)*b*c*e^2*log(F)*sin(d) + F^(a*c)*e^3*cos(d))*x)*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b^2*c^2*log(F)^2*sin(d
) - 2*F^(a*c)*b*c*e*cos(d)*log(F) - F^(a*c)*e^2*sin(d) - (F^(a*c)*b^3*c^3*log(F)^3*sin(d) - F^(a*c)*b^2*c^2*e*
cos(d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(d) - F^(a*c)*e^3*cos(d))*x)*F^(b*c*x)*sin(e*x))*e/(b^4*c^4*cos(d)
^2*log(F)^4 + b^4*c^4*log(F)^4*sin(d)^2 + (cos(d)^2 + sin(d)^2)*e^4 + 2*(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*l
og(F)^2*sin(d)^2)*e^2) - 1/2*((F^(a*c)*b*c*log(F)*sin(d) + F^(a*c)*e*cos(d))*F^(b*c*x)*cos(e*x + 2*d) - (F^(a*
c)*b*c*log(F)*sin(d) - F^(a*c)*e*cos(d))*F^(b*c*x)*cos(e*x) - (F^(a*c)*b*c*cos(d)*log(F) - F^(a*c)*e*sin(d))*F
^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b*c*cos(d)*log(F) + F^(a*c)*e*sin(d))*F^(b*c*x)*sin(e*x))/(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)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 1941, normalized size of antiderivative = 114.18 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 27.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} (e x \cos (d+e x)+(1+b c x \log (F)) \sin (d+e x)) \, dx=F^{c\,\left (a+b\,x\right )}\,x\,\sin \left (d+e\,x\right ) \]

[In]

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

[Out]

F^(c*(a + b*x))*x*sin(d + e*x)