3.3.87 \(\int \frac {(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [287]

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

[Out]

1/2*(f*x+e)^(1+m)/a/f/(1+m)+1/8*exp(I*(c-d*e/f))*(f*x+e)^m*GAMMA(1+m,-I*d*(f*x+e)/f)/a/d/((-I*d*(f*x+e)/f)^m)+
1/8*(f*x+e)^m*GAMMA(1+m,I*d*(f*x+e)/f)/a/d/exp(I*(c-d*e/f))/((I*d*(f*x+e)/f)^m)-I*2^(-3-m)*exp(2*I*(c-d*e/f))*
(f*x+e)^m*GAMMA(1+m,-2*I*d*(f*x+e)/f)/a/d/((-I*d*(f*x+e)/f)^m)+I*2^(-3-m)*(f*x+e)^m*GAMMA(1+m,2*I*d*(f*x+e)/f)
/a/d/exp(2*I*(c-d*e/f))/((I*d*(f*x+e)/f)^m)+1/8*3^(-1-m)*exp(3*I*(c-d*e/f))*(f*x+e)^m*GAMMA(1+m,-3*I*d*(f*x+e)
/f)/a/d/((-I*d*(f*x+e)/f)^m)+1/8*3^(-1-m)*(f*x+e)^m*GAMMA(1+m,3*I*d*(f*x+e)/f)/a/d/exp(3*I*(c-d*e/f))/((I*d*(f
*x+e)/f)^m)

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Rubi [A]
time = 0.42, antiderivative size = 449, normalized size of antiderivative = 1.00, 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 = {4619, 3393, 3388, 2212, 4491, 3389} \begin {gather*} \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)} \end {gather*}

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 2212

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

Rule 3388

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 3389

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]

Rule 3393

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 4491

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 4619

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]

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*}

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Mathematica [A]
time = 2.61, size = 411, normalized size = 0.92 \begin {gather*} \frac {i (e+f x)^m \left (-\frac {12 i d e}{f+f m}-\frac {12 i d x}{1+m}-3 i e^{i \left (c-\frac {d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {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} \Gamma \left (1+m,\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} \Gamma \left (1+m,-\frac {2 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} \Gamma \left (1+m,\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} \Gamma \left (1+m,-\frac {3 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} \Gamma \left (1+m,\frac {3 i d (e+f x)}{f}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}{24 a d (1+\sin (c+d x))} \end {gather*}

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 + f*m) - ((12*I)*d*x)/(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 = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{m} \left (\cos ^{4}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.12, size = 354, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {i \, d}{f}\right ) + i \, c f - i \, d e}{f}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, d e}{f}\right ) - 3 \, {\left (i \, f m + i \, f\right )} e^{\left (-\frac {f m \log \left (-\frac {2 i \, d}{f}\right ) - 2 i \, c f + 2 i \, d e}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {3 i \, d}{f}\right ) - 3 i \, c f + 3 i \, d e}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, d f x + i \, d e\right )}}{f}\right ) + 3 \, {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (-\frac {i \, d}{f}\right ) - i \, c f + i \, d e}{f}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, d e}{f}\right ) - 3 \, {\left (-i \, f m - i \, f\right )} e^{\left (-\frac {f m \log \left (\frac {2 i \, d}{f}\right ) + 2 i \, c f - 2 i \, d e}{f}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, d e\right )}}{f}\right ) + {\left (f m + f\right )} e^{\left (-\frac {f m \log \left (\frac {3 i \, d}{f}\right ) + 3 i \, c f - 3 i \, d e}{f}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, d f x - i \, d e\right )}}{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 {gather*}

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*(3*(f*m + f)*e^(-(f*m*log(I*d/f) + I*c*f - I*d*e)/f)*gamma(m + 1, (I*d*f*x + I*d*e)/f) - 3*(I*f*m + I*f)*
e^(-(f*m*log(-2*I*d/f) - 2*I*c*f + 2*I*d*e)/f)*gamma(m + 1, -2*(I*d*f*x + I*d*e)/f) + (f*m + f)*e^(-(f*m*log(-
3*I*d/f) - 3*I*c*f + 3*I*d*e)/f)*gamma(m + 1, -3*(I*d*f*x + I*d*e)/f) + 3*(f*m + f)*e^(-(f*m*log(-I*d/f) - I*c
*f + I*d*e)/f)*gamma(m + 1, (-I*d*f*x - I*d*e)/f) - 3*(-I*f*m - I*f)*e^(-(f*m*log(2*I*d/f) + 2*I*c*f - 2*I*d*e
)/f)*gamma(m + 1, -2*(-I*d*f*x - I*d*e)/f) + (f*m + f)*e^(-(f*m*log(3*I*d/f) + 3*I*c*f - 3*I*d*e)/f)*gamma(m +
1, -3*(-I*d*f*x - 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (e+f\,x\right )}^m}{a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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