3.33 \(\int \frac{F^{c (a+b x)} (f x)^m (e x \cos (d+e x)+(m+b c x \log (F)) \sin (d+e x))}{x} \, dx\)

Optimal. Leaf size=22 \[ (f x)^m \sin (d+e x) F^{a c+b c x} \]

[Out]

F^(a*c + b*c*x)*(f*x)^m*Sin[d + e*x]

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Rubi [A]  time = 2.5633, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {16, 6741, 6742, 4467} \[ (f x)^m \sin (d+e x) F^{a c+b c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

F^(a*c + b*c*x)*(f*x)^m*Sin[d + e*x]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 6741

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

Rule 6742

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

Rule 4467

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_)*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[((f*x)^(m +
1)*F^(c*(a + b*x))*Sin[d + e*x])/(f*(m + 1)), x] + (-Dist[e/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Co
s[d + e*x], x], x] - Dist[(b*c*Log[F])/(f*(m + 1)), Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x], x]) /;
FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])

Rubi steps

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

Mathematica [A]  time = 0.863564, size = 22, normalized size = 1. \[ (f x)^m \sin (d+e x) F^{a c+b c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

F^(a*c + b*c*x)*(f*x)^m*Sin[d + e*x]

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Maple [C]  time = 0.15, size = 199, normalized size = 9.1 \begin{align*} -{\frac{i}{2}}{F}^{c \left ( bx+a \right ) } \left ({x}^{m}{f}^{m}{{\rm e}^{iex}}{{\rm e}^{id}}{{\rm e}^{-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}m}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) m}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) m}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) m}}-{x}^{m}{f}^{m}{{\rm e}^{-iex}}{{\rm e}^{-id}}{{\rm e}^{-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}m}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) m}}{{\rm e}^{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) m}}{{\rm e}^{-{\frac{i}{2}}\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) m}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*F^(c*(b*x+a))*(x^m*f^m*exp(I*e*x)*exp(I*d)*exp(-1/2*I*Pi*csgn(I*f*x)^3*m)*exp(1/2*I*Pi*csgn(I*f*x)^2*cs
gn(I*f)*m)*exp(1/2*I*Pi*csgn(I*f*x)^2*csgn(I*x)*m)*exp(-1/2*I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)*m)-x^m*f^m*ex
p(-I*e*x)*exp(-I*d)*exp(-1/2*I*Pi*csgn(I*f*x)^3*m)*exp(1/2*I*Pi*csgn(I*f*x)^2*csgn(I*f)*m)*exp(1/2*I*Pi*csgn(I
*f*x)^2*csgn(I*x)*m)*exp(-1/2*I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)*m))

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Maxima [A]  time = 2.14592, size = 36, normalized size = 1.64 \begin{align*} F^{a c} f^{m} e^{\left (b c x \log \left (F\right ) + m \log \left (x\right )\right )} \sin \left (e x + d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.494112, size = 51, normalized size = 2.32 \begin{align*} \left (f x\right )^{m} F^{b c x + a c} \sin \left (e x + d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(f*x)^m*F^(b*c*x + a*c)*sin(e*x + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x \cos \left (e x + d\right ) +{\left (b c x \log \left (F\right ) + m\right )} \sin \left (e x + d\right )\right )} \left (f x\right )^{m} F^{{\left (b x + a\right )} c}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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