3.947 \(\int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx\)

Optimal. Leaf size=119 \[ -\frac {8 \sqrt {2} a (1-\sin (e+f x)) \cos ^3(e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {5}{2};-\frac {5}{2},-n;\frac {7}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{5 f (\sin (e+f x)+1)^{3/2}} \]

[Out]

-8/5*a*AppellF1(5/2,-n,-5/2,7/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*cos(f*x+e)^3*(1-sin(f*x+e))*(c+d*si
n(f*x+e))^n*2^(1/2)/f/(1+sin(f*x+e))^(3/2)/(((c+d*sin(f*x+e))/(c+d))^n)

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Rubi [A]  time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2868, 139, 138} \[ -\frac {8 \sqrt {2} a (1-\sin (e+f x)) \cos ^3(e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {5}{2};-\frac {5}{2},-n;\frac {7}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{5 f (\sin (e+f x)+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[2]*a*AppellF1[5/2, -5/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]^
3*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(5*f*(1 + Sin[e + f*x])^(3/2)*((c + d*Sin[e + f*x])/(c + d))^n)

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 139

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2868

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)]), x_Symbol] :> Dist[(c*g*(g*Cos[e + f*x])^(p - 1))/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e +
f*x])^((p - 1)/2)), Subst[Int[(1 + (d*x)/c)^((p + 1)/2)*(1 - (d*x)/c)^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e +
f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]

Rubi steps

\begin {align*} \int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx &=\frac {\left (a \cos ^3(e+f x)\right ) \operatorname {Subst}\left (\int (1-x)^{3/2} (1+x)^{5/2} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f (1-\sin (e+f x))^{3/2} (1+\sin (e+f x))^{3/2}}\\ &=\frac {\left (a \cos ^3(e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname {Subst}\left (\int (1-x)^{3/2} (1+x)^{5/2} \left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n \, dx,x,\sin (e+f x)\right )}{f (1-\sin (e+f x))^{3/2} (1+\sin (e+f x))^{3/2}}\\ &=-\frac {8 \sqrt {2} a F_1\left (\frac {5}{2};-\frac {5}{2},-n;\frac {7}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos ^3(e+f x) (1-\sin (e+f x)) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{5 f (1+\sin (e+f x))^{3/2}}\\ \end {align*}

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Mathematica [F]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) + a \cos \left (f x + e\right )^{4}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*cos(f*x + e)^4*sin(f*x + e) + a*cos(f*x + e)^4)*(d*sin(f*x + e) + c)^n, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.56, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (e+f\,x\right )}^4\,\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**4*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))**n,x)

[Out]

Timed out

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