3.13 \(\int F^{c (a+b x)} \cos (d+e x) \, dx\)

Optimal. Leaf size=72 \[ \frac{e \sin (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}+\frac{b c \log (F) \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2} \]

[Out]

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

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Rubi [A]  time = 0.0161595, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4433} \[ \frac{e \sin (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}+\frac{b c \log (F) \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Cos[d + e*x],x]

[Out]

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

Rule 4433

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]

Rubi steps

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

Mathematica [A]  time = 0.0965807, size = 47, normalized size = 0.65 \[ \frac{F^{c (a+b x)} (b c \log (F) \cos (d+e x)+e \sin (d+e x))}{b^2 c^2 \log ^2(F)+e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 133, normalized size = 1.9 \begin{align*}{ \left ({\frac{bc\ln \left ( F \right ){{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}}{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}+2\,{\frac{e{{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}\tan \left ( d/2+1/2\,ex \right ) }{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}-{\frac{bc\ln \left ( F \right ){{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}}{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln(F)*b*c/(e^2+b^2*c^2*ln(F)^2)*exp(c*(b*x+a)*ln(F))+2/(e^2+b^2*c^2*ln(F)^2)*e*exp(c*(b*x+a)*ln(F))*tan(1/2*d
+1/2*e*x)-ln(F)*b*c/(e^2+b^2*c^2*ln(F)^2)*exp(c*(b*x+a)*ln(F))*tan(1/2*d+1/2*e*x)^2)/(1+tan(1/2*d+1/2*e*x)^2)

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Maxima [B]  time = 1.08046, size = 259, normalized size = 3.6 \begin{align*} \frac{{\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 )}{2 \,{\left (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}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((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))/(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)

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Fricas [A]  time = 0.476363, size = 115, normalized size = 1.6 \begin{align*} \frac{{\left (b c \cos \left (e x + d\right ) \log \left (F\right ) + e \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{2} c^{2} \log \left (F\right )^{2} + e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 46.2778, size = 352, normalized size = 4.89 \begin{align*} \begin{cases} - \frac{\left (-1\right )^{a c} \left (-1\right )^{\frac{e x}{\pi }} i x \sin{\left (d + e x \right )}}{2} + \frac{\left (-1\right )^{a c} \left (-1\right )^{\frac{e x}{\pi }} x \cos{\left (d + e x \right )}}{2} + \frac{\left (-1\right )^{a c} \left (-1\right )^{\frac{e x}{\pi }} \sin{\left (d + e x \right )}}{e} + \frac{\left (-1\right )^{a c} \left (-1\right )^{\frac{e x}{\pi }} i \cos{\left (d + e x \right )}}{2 e} & \text{for}\: F = -1 \wedge b = \frac{e}{\pi c} \\x \cos{\left (d \right )} & \text{for}\: F = 1 \wedge e = 0 \\\tilde{\infty } e \left (e^{- \frac{i e}{b c}}\right )^{a c} \left (e^{- \frac{i e}{b c}}\right )^{b c x} \sin{\left (d + e x \right )} + \tilde{\infty } e \left (e^{- \frac{i e}{b c}}\right )^{a c} \left (e^{- \frac{i e}{b c}}\right )^{b c x} \cos{\left (d + e x \right )} & \text{for}\: F = e^{- \frac{i e}{b c}} \\\tilde{\infty } e \left (e^{\frac{i e}{b c}}\right )^{a c} \left (e^{\frac{i e}{b c}}\right )^{b c x} \sin{\left (d + e x \right )} + \tilde{\infty } e \left (e^{\frac{i e}{b c}}\right )^{a c} \left (e^{\frac{i e}{b c}}\right )^{b c x} \cos{\left (d + e x \right )} & \text{for}\: F = e^{\frac{i e}{b c}} \\\frac{F^{a c} F^{b c x} b c \log{\left (F \right )} \cos{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} + e^{2}} + \frac{F^{a c} F^{b c x} e \sin{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} + e^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(-1)**(a*c)*(-1)**(e*x/pi)*I*x*sin(d + e*x)/2 + (-1)**(a*c)*(-1)**(e*x/pi)*x*cos(d + e*x)/2 + (-1)
**(a*c)*(-1)**(e*x/pi)*sin(d + e*x)/e + (-1)**(a*c)*(-1)**(e*x/pi)*I*cos(d + e*x)/(2*e), Eq(F, -1) & Eq(b, e/(
pi*c))), (x*cos(d), Eq(F, 1) & Eq(e, 0)), (zoo*e*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*sin(d + e*x)
+ zoo*e*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*cos(d + e*x), Eq(F, exp(-I*e/(b*c)))), (zoo*e*exp(I*e/
(b*c))**(a*c)*exp(I*e/(b*c))**(b*c*x)*sin(d + e*x) + zoo*e*exp(I*e/(b*c))**(a*c)*exp(I*e/(b*c))**(b*c*x)*cos(d
+ e*x), Eq(F, exp(I*e/(b*c)))), (F**(a*c)*F**(b*c*x)*b*c*log(F)*cos(d + e*x)/(b**2*c**2*log(F)**2 + e**2) + F
**(a*c)*F**(b*c*x)*e*sin(d + e*x)/(b**2*c**2*log(F)**2 + e**2), True))

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Giac [C]  time = 1.19659, size = 876, normalized size = 12.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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