3.287 \(\int \frac{(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=449 \[ \frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{8 a d}-\frac{i 2^{-m-3} e^{2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{3^{-m-1} e^{3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i d (e+f x)}{f}\right )}{8 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{8 a d}+\frac{i 2^{-m-3} e^{-2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{3^{-m-1} e^{-3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i d (e+f x)}{f}\right )}{8 a d}+\frac{(e+f x)^{m+1}}{2 a f (m+1)} \]

[Out]

(e + f*x)^(1 + m)/(2*a*f*(1 + m)) + (E^(I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f])/(8*a*
d*(((-I)*d*(e + f*x))/f)^m) + ((e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(8*a*d*E^(I*(c - (d*e)/f))*((I*d*(
e + f*x))/f)^m) - (I*2^(-3 - m)*E^((2*I)*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f])/(a*d
*(((-I)*d*(e + f*x))/f)^m) + (I*2^(-3 - m)*(e + f*x)^m*Gamma[1 + m, ((2*I)*d*(e + f*x))/f])/(a*d*E^((2*I)*(c -
(d*e)/f))*((I*d*(e + f*x))/f)^m) + (3^(-1 - m)*E^((3*I)*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-3*I)*d*(e
+ f*x))/f])/(8*a*d*(((-I)*d*(e + f*x))/f)^m) + (3^(-1 - m)*(e + f*x)^m*Gamma[1 + m, ((3*I)*d*(e + f*x))/f])/(8
*a*d*E^((3*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m)

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Rubi [A]  time = 0.642543, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4523, 3312, 3307, 2181, 4406, 3308} \[ \frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{8 a d}-\frac{i 2^{-m-3} e^{2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{3^{-m-1} e^{3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i d (e+f x)}{f}\right )}{8 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{8 a d}+\frac{i 2^{-m-3} e^{-2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{3^{-m-1} e^{-3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i d (e+f x)}{f}\right )}{8 a d}+\frac{(e+f x)^{m+1}}{2 a f (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^m*Cos[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(e + f*x)^(1 + m)/(2*a*f*(1 + m)) + (E^(I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f])/(8*a*
d*(((-I)*d*(e + f*x))/f)^m) + ((e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(8*a*d*E^(I*(c - (d*e)/f))*((I*d*(
e + f*x))/f)^m) - (I*2^(-3 - m)*E^((2*I)*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f])/(a*d
*(((-I)*d*(e + f*x))/f)^m) + (I*2^(-3 - m)*(e + f*x)^m*Gamma[1 + m, ((2*I)*d*(e + f*x))/f])/(a*d*E^((2*I)*(c -
(d*e)/f))*((I*d*(e + f*x))/f)^m) + (3^(-1 - m)*E^((3*I)*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, ((-3*I)*d*(e
+ f*x))/f])/(8*a*d*(((-I)*d*(e + f*x))/f)^m) + (3^(-1 - m)*(e + f*x)^m*Gamma[1 + m, ((3*I)*d*(e + f*x))/f])/(8
*a*d*E^((3*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m)

Rule 4523

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*S
in[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rubi steps

\begin{align*} \int \frac{(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x)^m \cos ^2(c+d x) \, dx}{a}-\frac{\int (e+f x)^m \cos ^2(c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac{\int \left (\frac{1}{2} (e+f x)^m+\frac{1}{2} (e+f x)^m \cos (2 c+2 d x)\right ) \, dx}{a}-\frac{\int \left (\frac{1}{4} (e+f x)^m \sin (c+d x)+\frac{1}{4} (e+f x)^m \sin (3 c+3 d x)\right ) \, dx}{a}\\ &=\frac{(e+f x)^{1+m}}{2 a f (1+m)}-\frac{\int (e+f x)^m \sin (c+d x) \, dx}{4 a}-\frac{\int (e+f x)^m \sin (3 c+3 d x) \, dx}{4 a}+\frac{\int (e+f x)^m \cos (2 c+2 d x) \, dx}{2 a}\\ &=\frac{(e+f x)^{1+m}}{2 a f (1+m)}-\frac{i \int e^{-i (c+d x)} (e+f x)^m \, dx}{8 a}+\frac{i \int e^{i (c+d x)} (e+f x)^m \, dx}{8 a}-\frac{i \int e^{-i (3 c+3 d x)} (e+f x)^m \, dx}{8 a}+\frac{i \int e^{i (3 c+3 d x)} (e+f x)^m \, dx}{8 a}+\frac{\int e^{-i (2 c+2 d x)} (e+f x)^m \, dx}{4 a}+\frac{\int e^{i (2 c+2 d x)} (e+f x)^m \, dx}{4 a}\\ &=\frac{(e+f x)^{1+m}}{2 a f (1+m)}+\frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac{i d (e+f x)}{f}\right )}{8 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac{i d (e+f x)}{f}\right )}{8 a d}-\frac{i 2^{-3-m} e^{2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{i 2^{-3-m} e^{-2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{3^{-1-m} e^{3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac{3 i d (e+f x)}{f}\right )}{8 a d}+\frac{3^{-1-m} e^{-3 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac{3 i d (e+f x)}{f}\right )}{8 a d}\\ \end{align*}

Mathematica [A]  time = 4.71712, size = 405, normalized size = 0.9 \[ \frac{i (e+f x)^m \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (-3 i e^{i \left (c-\frac{d e}{f}\right )} \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )-3\ 2^{-m} e^{2 i \left (c-\frac{d e}{f}\right )} \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )-i 3^{-m} e^{3 i \left (c-\frac{d e}{f}\right )} \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i d (e+f x)}{f}\right )-3 i e^{-i \left (c-\frac{d e}{f}\right )} \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )+3\ 2^{-m} e^{-2 i \left (c-\frac{d e}{f}\right )} \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )-i 3^{-m} e^{-3 i \left (c-\frac{d e}{f}\right )} \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i d (e+f x)}{f}\right )-\frac{12 i d (e+f x)}{f (m+1)}\right )}{24 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^m*Cos[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

((I/24)*(e + f*x)^m*(((-12*I)*d*(e + f*x))/(f*(1 + m)) - ((3*I)*E^(I*(c - (d*e)/f))*Gamma[1 + m, ((-I)*d*(e +
f*x))/f])/(((-I)*d*(e + f*x))/f)^m - ((3*I)*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f
*x))/f)^m) - (3*E^((2*I)*(c - (d*e)/f))*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f])/(2^m*(((-I)*d*(e + f*x))/f)^m) +
(3*Gamma[1 + m, ((2*I)*d*(e + f*x))/f])/(2^m*E^((2*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m) - (I*E^((3*I)*(c
- (d*e)/f))*Gamma[1 + m, ((-3*I)*d*(e + f*x))/f])/(3^m*(((-I)*d*(e + f*x))/f)^m) - (I*Gamma[1 + m, ((3*I)*d*(e
+ f*x))/f])/(3^m*E^((3*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(a*
d*(1 + Sin[c + d*x]))

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{a+a\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^m*cos(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

int((f*x+e)^m*cos(d*x+c)^4/(a+a*sin(d*x+c)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*cos(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate((f*x + e)^m*cos(d*x + c)^4/(a*sin(d*x + c) + a), x)

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Fricas [A]  time = 1.99043, size = 861, normalized size = 1.92 \begin{align*} \frac{{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (\frac{3 i \, d}{f}\right ) - 3 i \, d e + 3 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{3 i \, d f x + 3 i \, d e}{f}\right ) +{\left (3 i \, f m + 3 i \, f\right )} e^{\left (-\frac{f m \log \left (\frac{2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{2 i \, d f x + 2 i \, d e}{f}\right ) + 3 \,{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (\frac{i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{i \, d f x + i \, d e}{f}\right ) + 3 \,{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (-\frac{i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{-i \, d f x - i \, d e}{f}\right ) +{\left (-3 i \, f m - 3 i \, f\right )} e^{\left (-\frac{f m \log \left (-\frac{2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{-2 i \, d f x - 2 i \, d e}{f}\right ) +{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (-\frac{3 i \, d}{f}\right ) + 3 i \, d e - 3 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{-3 i \, d f x - 3 i \, d e}{f}\right ) + 12 \,{\left (d f x + d e\right )}{\left (f x + e\right )}^{m}}{24 \,{\left (a d f m + a d f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*cos(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/24*((f*m + f)*e^(-(f*m*log(3*I*d/f) - 3*I*d*e + 3*I*c*f)/f)*gamma(m + 1, (3*I*d*f*x + 3*I*d*e)/f) + (3*I*f*m
+ 3*I*f)*e^(-(f*m*log(2*I*d/f) - 2*I*d*e + 2*I*c*f)/f)*gamma(m + 1, (2*I*d*f*x + 2*I*d*e)/f) + 3*(f*m + f)*e^
(-(f*m*log(I*d/f) - I*d*e + I*c*f)/f)*gamma(m + 1, (I*d*f*x + I*d*e)/f) + 3*(f*m + f)*e^(-(f*m*log(-I*d/f) + I
*d*e - I*c*f)/f)*gamma(m + 1, (-I*d*f*x - I*d*e)/f) + (-3*I*f*m - 3*I*f)*e^(-(f*m*log(-2*I*d/f) + 2*I*d*e - 2*
I*c*f)/f)*gamma(m + 1, (-2*I*d*f*x - 2*I*d*e)/f) + (f*m + f)*e^(-(f*m*log(-3*I*d/f) + 3*I*d*e - 3*I*c*f)/f)*ga
mma(m + 1, (-3*I*d*f*x - 3*I*d*e)/f) + 12*(d*f*x + d*e)*(f*x + e)^m)/(a*d*f*m + a*d*f)

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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*x+e)**m*cos(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*cos(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^m*cos(d*x + c)^4/(a*sin(d*x + c) + a), x)