3.1021 \(\int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=123 \[ \frac {B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}+\frac {(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}-\frac {4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac {4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)} \]

[Out]

4*(A-B)*(a+a*sin(f*x+e))^(3+m)/a^3/f/(3+m)-4*(A-2*B)*(a+a*sin(f*x+e))^(4+m)/a^4/f/(4+m)+(A-5*B)*(a+a*sin(f*x+e
))^(5+m)/a^5/f/(5+m)+B*(a+a*sin(f*x+e))^(6+m)/a^6/f/(6+m)

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Rubi [A]  time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 77} \[ \frac {4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac {4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac {(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac {B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

(4*(A - B)*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) - (4*(A - 2*B)*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(
4 + m)) + ((A - 5*B)*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m)) + (B*(a + a*Sin[e + f*x])^(6 + m))/(a^6*f*(
6 + m))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2836

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

Rubi steps

\begin {align*} \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{2+m} \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^5 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^{2+m}-4 a (A-2 B) (a+x)^{3+m}+(A-5 B) (a+x)^{4+m}+\frac {B (a+x)^{5+m}}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^5 f}\\ &=\frac {4 (A-B) (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}-\frac {4 (A-2 B) (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac {(A-5 B) (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}+\frac {B (a+a \sin (e+f x))^{6+m}}{a^6 f (6+m)}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 103, normalized size = 0.84 \[ \frac {(a (\sin (e+f x)+1))^{m+3} \left (\frac {a^3 (A-5 B) (\sin (e+f x)+1)^2}{m+5}-\frac {4 a^3 (A-2 B) (\sin (e+f x)+1)}{m+4}+\frac {4 a^3 (A-B)}{m+3}+\frac {B (a \sin (e+f x)+a)^3}{m+6}\right )}{a^6 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

((a*(1 + Sin[e + f*x]))^(3 + m)*((4*a^3*(A - B))/(3 + m) - (4*a^3*(A - 2*B)*(1 + Sin[e + f*x]))/(4 + m) + (a^3
*(A - 5*B)*(1 + Sin[e + f*x])^2)/(5 + m) + (B*(a + a*Sin[e + f*x])^3)/(6 + m)))/(a^6*f)

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fricas [A]  time = 0.66, size = 221, normalized size = 1.80 \[ -\frac {{\left ({\left (B m^{3} + 12 \, B m^{2} + 47 \, B m + 60 \, B\right )} \cos \left (f x + e\right )^{6} - {\left ({\left (A + B\right )} m^{3} + 3 \, {\left (3 \, A + B\right )} m^{2} + 18 \, A m\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left ({\left (A + B\right )} m^{2} + 6 \, A m\right )} \cos \left (f x + e\right )^{2} - 32 \, {\left (A + B\right )} m - {\left ({\left ({\left (A + B\right )} m^{3} + {\left (13 \, A + 7 \, B\right )} m^{2} + 6 \, {\left (9 \, A + 2 \, B\right )} m + 72 \, A\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left ({\left (A + B\right )} m^{2} + 2 \, {\left (4 \, A + B\right )} m + 12 \, A\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (A + B\right )} m + 192 \, A\right )} \sin \left (f x + e\right ) - 192 \, A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{4} + 18 \, f m^{3} + 119 \, f m^{2} + 342 \, f m + 360 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

-((B*m^3 + 12*B*m^2 + 47*B*m + 60*B)*cos(f*x + e)^6 - ((A + B)*m^3 + 3*(3*A + B)*m^2 + 18*A*m)*cos(f*x + e)^4
- 8*((A + B)*m^2 + 6*A*m)*cos(f*x + e)^2 - 32*(A + B)*m - (((A + B)*m^3 + (13*A + 7*B)*m^2 + 6*(9*A + 2*B)*m +
72*A)*cos(f*x + e)^4 + 8*((A + B)*m^2 + 2*(4*A + B)*m + 12*A)*cos(f*x + e)^2 + 32*(A + B)*m + 192*A)*sin(f*x
+ e) - 192*A)*(a*sin(f*x + e) + a)^m/(f*m^4 + 18*f*m^3 + 119*f*m^2 + 342*f*m + 360*f)

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giac [B]  time = 0.23, size = 861, normalized size = 7.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

(((a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m^2 - 4*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m^2 +
4*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m^2 + 7*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m -
32*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m + 36*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m
+ 12*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m - 60*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a + 80*(
a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2)*A/(a^4*m^3 + 12*a^4*m^2 + 47*a^4*m + 60*a^4) + ((a*sin(f*x +
e) + a)^6*(a*sin(f*x + e) + a)^m*m^3 - 5*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m^3 + 8*(a*sin(f*x +
e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m^3 - 4*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^3*m^3 + 12*(a*si
n(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*m^2 - 65*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m^2 + 112*(a
*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m^2 - 60*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^3*m^2
+ 47*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*m - 270*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m
+ 504*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m - 296*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*
a^3*m + 60*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m - 360*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a
+ 720*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2 - 480*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a
^3)*B/((a^4*m^4 + 18*a^4*m^3 + 119*a^4*m^2 + 342*a^4*m + 360*a^4)*a))/(a*f)

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maple [F]  time = 13.37, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^5*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

[Out]

int(cos(f*x+e)^5*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

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maxima [B]  time = 0.42, size = 643, normalized size = 5.23 \[ \frac {\frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (f x + e\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 24 \, a^{m} m \sin \left (f x + e\right ) + 24 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} a^{m} \sin \left (f x + e\right )^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (f x + e\right )^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 60 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 120 \, a^{m} m \sin \left (f x + e\right ) - 120 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} - \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} A}{a {\left (m + 1\right )}}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

(((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*a^m*sin(f*x + e)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + e)^4 - 4
*(m^3 + 3*m^2 + 2*m)*a^m*sin(f*x + e)^3 + 12*(m^2 + m)*a^m*sin(f*x + e)^2 - 24*a^m*m*sin(f*x + e) + 24*a^m)*A*
(sin(f*x + e) + 1)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) - 2*((m^2 + 3*m + 2)*a^m*sin(f*x + e)^3 +
(m^2 + m)*a^m*sin(f*x + e)^2 - 2*a^m*m*sin(f*x + e) + 2*a^m)*A*(sin(f*x + e) + 1)^m/(m^3 + 6*m^2 + 11*m + 6)
+ ((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*a^m*sin(f*x + e)^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m
)*a^m*sin(f*x + e)^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + e)^4 + 20*(m^3 + 3*m^2 + 2*m)*a^m*sin(f*x
+ e)^3 - 60*(m^2 + m)*a^m*sin(f*x + e)^2 + 120*a^m*m*sin(f*x + e) - 120*a^m)*B*(sin(f*x + e) + 1)^m/(m^6 + 21*
m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) - 2*((m^3 + 6*m^2 + 11*m + 6)*a^m*sin(f*x + e)^4 + (m^3 + 3
*m^2 + 2*m)*a^m*sin(f*x + e)^3 - 3*(m^2 + m)*a^m*sin(f*x + e)^2 + 6*a^m*m*sin(f*x + e) - 6*a^m)*B*(sin(f*x + e
) + 1)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + (a^m*(m + 1)*sin(f*x + e)^2 + a^m*m*sin(f*x + e) - a^m)*B*(sin(
f*x + e) + 1)^m/(m^2 + 3*m + 2) + (a*sin(f*x + e) + a)^(m + 1)*A/(a*(m + 1)))/f

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mupad [B]  time = 15.82, size = 517, normalized size = 4.20 \[ -{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (12288\,A-1200\,B+4016\,A\,m+1108\,B\,m+472\,A\,m^2+24\,A\,m^3+88\,B\,m^2+4\,B\,m^3\right )}{64\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (1056\,A\,m-900\,B-705\,B\,m+272\,A\,m^2+16\,A\,m^3-4\,B\,m^2+B\,m^3\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\left (m+3\right )\,\left (60\,B-12\,A\,m+27\,B\,m-2\,A\,m^2+B\,m^2\right )}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+23\,m+300\right )\,1{}\mathrm {i}}{8\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+7\,m+12\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {B\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\left (m^3+12\,m^2+47\,m+60\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (3\,m^2+53\,m+100\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)^5*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m,x)

[Out]

-exp(- e*6i - f*x*6i)*(a + a*sin(e + f*x))^m*((exp(e*6i + f*x*6i)*cos(4*e + 4*f*x)*(m + 3)*(60*B - 12*A*m + 27
*B*m - 2*A*m^2 + B*m^2))/(16*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) - (exp(e*6i + f*x*6i)*cos(2*e + 2*f*x)*
(1056*A*m - 900*B - 705*B*m + 272*A*m^2 + 16*A*m^3 - 4*B*m^2 + B*m^3))/(32*f*(342*m + 119*m^2 + 18*m^3 + m^4 +
360)) - (exp(e*6i + f*x*6i)*(12288*A - 1200*B + 4016*A*m + 1108*B*m + 472*A*m^2 + 24*A*m^3 + 88*B*m^2 + 4*B*m
^3))/(64*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f*x*6i)*sin(e + f*x)*(A*6i + A*m*1i + B*m*1i)
*(23*m + m^2 + 300)*1i)/(8*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f*x*6i)*sin(5*e + 5*f*x)*(A
*6i + A*m*1i + B*m*1i)*(7*m + m^2 + 12)*1i)/(16*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (B*exp(e*6i + f*x*
6i)*cos(6*e + 6*f*x)*(47*m + 12*m^2 + m^3 + 60))/(32*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f
*x*6i)*sin(3*e + 3*f*x)*(A*6i + A*m*1i + B*m*1i)*(53*m + 3*m^2 + 100)*1i)/(16*f*(342*m + 119*m^2 + 18*m^3 + m^
4 + 360)))

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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)**5*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)

[Out]

Timed out

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