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

Optimal. Leaf size=139 \[ -\frac{e^{i d} F^{a c} (f x)^m (-x (b c \log (F)+i e))^{-m} \text{Gamma}(m+1,-x (b c \log (F)+i e))}{2 (e-i b c \log (F))}-\frac{e^{-i d} F^{a c} (f x)^m (x (-b c \log (F)+i e))^{-m} \text{Gamma}(m+1,x (-b c \log (F)+i e))}{2 (e+i b c \log (F))} \]

[Out]

-(F^(a*c)*(f*x)^m*Gamma[1 + m, x*(I*e - b*c*Log[F])])/(2*E^(I*d)*(x*(I*e - b*c*Log[F]))^m*(e + I*b*c*Log[F]))
- (E^(I*d)*F^(a*c)*(f*x)^m*Gamma[1 + m, -(x*(I*e + b*c*Log[F]))])/(2*(e - I*b*c*Log[F])*(-(x*(I*e + b*c*Log[F]
)))^m)

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Rubi [F]  time = 0.490977, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int F^{c (a+b x)} (f x)^m \sin (d+e x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.578081, size = 143, normalized size = 1.03 \[ \frac{1}{2} F^{a c} (f x)^m (-x (b c \log (F)+i e))^{-m} \left (-i x (\cos (d)-i \sin (d)) (-b c x \log (F)-i e x)^m (i x (e+i b c \log (F)))^{-m-1} \text{Gamma}(m+1,-b c x \log (F)+i e x)-\frac{(\cos (d)+i \sin (d)) \text{Gamma}(m+1,-x (b c \log (F)+i e))}{e-i b c \log (F)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a*c)*(f*x)^m*((-I)*x*Gamma[1 + m, I*e*x - b*c*x*Log[F]]*(I*x*(e + I*b*c*Log[F]))^(-1 - m)*((-I)*e*x - b*c*
x*Log[F])^m*(Cos[d] - I*Sin[d]) - (Gamma[1 + m, -(x*(I*e + b*c*Log[F]))]*(Cos[d] + I*Sin[d]))/(e - I*b*c*Log[F
])))/(2*(-(x*(I*e + b*c*Log[F])))^m)

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Maple [F]  time = 0.366, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( fx \right ) ^{m}\sin \left ( ex+d \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.519904, size = 336, normalized size = 2.42 \begin{align*} \frac{{\left (i \, b c \log \left (F\right ) - e\right )} e^{\left (a c \log \left (F\right ) - m \log \left (-\frac{b c \log \left (F\right ) - i \, e}{f}\right ) - i \, d\right )} \Gamma \left (m + 1, -b c x \log \left (F\right ) + i \, e x\right ) +{\left (-i \, b c \log \left (F\right ) - e\right )} e^{\left (a c \log \left (F\right ) - m \log \left (-\frac{b c \log \left (F\right ) + i \, e}{f}\right ) + i \, d\right )} \Gamma \left (m + 1, -b c x \log \left (F\right ) - i \, e x\right )}{2 \,{\left (b^{2} c^{2} \log \left (F\right )^{2} + e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((I*b*c*log(F) - e)*e^(a*c*log(F) - m*log(-(b*c*log(F) - I*e)/f) - I*d)*gamma(m + 1, -b*c*x*log(F) + I*e*x
) + (-I*b*c*log(F) - e)*e^(a*c*log(F) - m*log(-(b*c*log(F) + I*e)/f) + I*d)*gamma(m + 1, -b*c*x*log(F) - I*e*x
))/(b^2*c^2*log(F)^2 + e^2)

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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*sin(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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