3.3.88 \(\int \frac {(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [288]

Optimal. Leaf size=277 \[ -\frac {i e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {i e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {2^{-3-m} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {2^{-3-m} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )}{a d} \]

[Out]

-1/2*I*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)+1/2*I*(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)+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)+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)

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4619, 3388, 2212, 4491, 12, 3389} \begin {gather*} -\frac {i 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 )}{2 a d}+\frac {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 {i 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 )}{2 a d}+\frac {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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^m*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

________________________________________________________________________________________

Mathematica [A]
time = 1.21, size = 253, normalized size = 0.91 \begin {gather*} \frac {2^{-3-m} e^{-\frac {2 i (d e+c f)}{f}} (e+f x)^m \left (\frac {d^2 (e+f x)^2}{f^2}\right )^{-m} \left (-i 2^{2+m} e^{i \left (3 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 )+i 2^{2+m} e^{i \left (c+\frac {3 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 )+e^{4 i c} \left (\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,-\frac {2 i d (e+f x)}{f}\right )+e^{\frac {4 i d e}{f}} \left (-\frac {i d (e+f x)}{f}\right )^m \Gamma \left (1+m,\frac {2 i d (e+f x)}{f}\right )\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^m*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(2^(-3 - m)*(e + f*x)^m*((-I)*2^(2 + m)*E^(I*(3*c + (d*e)/f))*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-I)*d*(e +
f*x))/f] + I*2^(2 + m)*E^(I*(c + (3*d*e)/f))*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, (I*d*(e + f*x))/f] + E^((4*
I)*c)*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f] + E^(((4*I)*d*e)/f)*(((-I)*d*(e + f*x))/f)^m*
Gamma[1 + m, ((2*I)*d*(e + f*x))/f]))/(a*d*E^(((2*I)*(d*e + c*f))/f)*((d^2*(e + f*x)^2)/f^2)^m)

________________________________________________________________________________________

Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{m} \left (\cos ^{3}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

int((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)

________________________________________________________________________________________

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)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate((f*x + e)^m*cos(d*x + c)^3/(a*sin(d*x + c) + a), x)

________________________________________________________________________________________

Fricas [A]
time = 0.11, size = 197, normalized size = 0.71 \begin {gather*} \frac {4 i \, 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 ) + 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 ) - 4 i \, 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 ) + 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 )}{8 \, a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8*(4*I*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) + 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) - 4*I*e^(-(f*m*log(-I*d/f) - I*c*f + I*d*e)/f)*gamm
a(m + 1, (-I*d*f*x - I*d*e)/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))/(a*d)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**m*cos(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^m*cos(d*x + c)^3/(a*sin(d*x + c) + a), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^3\,{\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)^3*(e + f*x)^m)/(a + a*sin(c + d*x)),x)

[Out]

int((cos(c + d*x)^3*(e + f*x)^m)/(a + a*sin(c + d*x)), x)

________________________________________________________________________________________