3.452 \(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) \, dx\)

Optimal. Leaf size=287 \[ -\frac{\sin (e+f x) \left (a^2 A (m+2)+2 a b B (m+1)+A b^2 (m+1)\right ) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left (a (m+3) (a B+2 A b)+b^2 B (m+2)\right ) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+4)+A b (m+3)) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+3)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x)) (c \cos (e+f x))^{m+1}}{c f (m+3)} \]

[Out]

(b*(A*b*(3 + m) + a*B*(4 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(3 + m)) + (b*B*(c*Cos[e +
f*x])^(1 + m)*(a + b*Cos[e + f*x])*Sin[e + f*x])/(c*f*(3 + m)) - ((A*b^2*(1 + m) + 2*a*b*B*(1 + m) + a^2*A*(2
+ m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f
*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^2*B*(2 + m) + a*(2*A*b + a*B)*(3 + m))*(c*Cos[e + f*x])^(2 + m)*H
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e +
f*x]^2])

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Rubi [A]  time = 0.541114, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2990, 3023, 2748, 2643} \[ -\frac{\sin (e+f x) \left (a^2 A (m+2)+2 a b B (m+1)+A b^2 (m+1)\right ) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left (a (m+3) (a B+2 A b)+b^2 B (m+2)\right ) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+4)+A b (m+3)) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+3)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x)) (c \cos (e+f x))^{m+1}}{c f (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x]),x]

[Out]

(b*(A*b*(3 + m) + a*B*(4 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(3 + m)) + (b*B*(c*Cos[e +
f*x])^(1 + m)*(a + b*Cos[e + f*x])*Sin[e + f*x])/(c*f*(3 + m)) - ((A*b^2*(1 + m) + 2*a*b*B*(1 + m) + a^2*A*(2
+ m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f
*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^2*B*(2 + m) + a*(2*A*b + a*B)*(3 + m))*(c*Cos[e + f*x])^(2 + m)*H
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e +
f*x]^2])

Rule 2990

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x
])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c -
b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m,
1] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (c \cos (e+f x))^m (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) \, dx &=\frac{b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x)) \sin (e+f x)}{c f (3+m)}+\frac{\int (c \cos (e+f x))^m \left (a c (b B (1+m)+a A (3+m))+c \left (b^2 B (2+m)+a (2 A b+a B) (3+m)\right ) \cos (e+f x)+b c (A b (3+m)+a B (4+m)) \cos ^2(e+f x)\right ) \, dx}{c (3+m)}\\ &=\frac{b (A b (3+m)+a B (4+m)) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (3+m)}+\frac{b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x)) \sin (e+f x)}{c f (3+m)}+\frac{\int (c \cos (e+f x))^m \left (c^2 (a (2+m) (b B (1+m)+a A (3+m))+b (1+m) (A b (3+m)+a B (4+m)))+c^2 (2+m) \left (b^2 B (2+m)+a (2 A b+a B) (3+m)\right ) \cos (e+f x)\right ) \, dx}{c^2 (2+m) (3+m)}\\ &=\frac{b (A b (3+m)+a B (4+m)) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (3+m)}+\frac{b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x)) \sin (e+f x)}{c f (3+m)}+\frac{\left (A b^2 (1+m)+2 a b B (1+m)+a^2 A (2+m)\right ) \int (c \cos (e+f x))^m \, dx}{2+m}+\frac{\left (b^2 B (2+m)+a (2 A b+a B) (3+m)\right ) \int (c \cos (e+f x))^{1+m} \, dx}{c (3+m)}\\ &=\frac{b (A b (3+m)+a B (4+m)) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (3+m)}+\frac{b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x)) \sin (e+f x)}{c f (3+m)}-\frac{\left (A b^2 (1+m)+2 a b B (1+m)+a^2 A (2+m)\right ) (c \cos (e+f x))^{1+m} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{c f (1+m) (2+m) \sqrt{\sin ^2(e+f x)}}-\frac{\left (b^2 B (2+m)+a (2 A b+a B) (3+m)\right ) (c \cos (e+f x))^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{c^2 f (2+m) (3+m) \sqrt{\sin ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.74602, size = 217, normalized size = 0.76 \[ \frac{\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left (\cos (e+f x) \left (b \cos (e+f x) \left (-\frac{(2 a B+A b) \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right )}{m+3}-\frac{b B \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(e+f x)\right )}{m+4}\right )-\frac{a (a B+2 A b) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right )}{m+2}\right )-\frac{a^2 A \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{m+1}\right )}{f \sqrt{\sin ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x]),x]

[Out]

(Cos[e + f*x]*(c*Cos[e + f*x])^m*(-((a^2*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(1 +
m)) + Cos[e + f*x]*(-((a*(2*A*b + a*B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2])/(2 + m))
+ b*Cos[e + f*x]*(-(((A*b + 2*a*B)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2])/(3 + m)) - (b
*B*Cos[e + f*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[e + f*x]^2])/(4 + m))))*Sin[e + f*x])/(f*Sqrt
[Sin[e + f*x]^2])

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Maple [F]  time = 1.823, size = 0, normalized size = 0. \begin{align*} \int \left ( c\cos \left ( fx+e \right ) \right ) ^{m} \left ( a+b\cos \left ( fx+e \right ) \right ) ^{2} \left ( A+B\cos \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^2*(A+B*cos(f*x+e)),x)

[Out]

int((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^2*(A+B*cos(f*x+e)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )}^{2} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^2*(A+B*cos(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^2*(c*cos(f*x + e))^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (f x + e\right )^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (f x + e\right )\right )} \left (c \cos \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^2*(A+B*cos(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*b^2*cos(f*x + e)^3 + A*a^2 + (2*B*a*b + A*b^2)*cos(f*x + e)^2 + (B*a^2 + 2*A*a*b)*cos(f*x + e))*(c
*cos(f*x + e))^m, x)

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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((c*cos(f*x+e))**m*(a+b*cos(f*x+e))**2*(A+B*cos(f*x+e)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (f x + e\right ) + A\right )}{\left (b \cos \left (f x + e\right ) + a\right )}^{2} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^2*(A+B*cos(f*x+e)),x, algorithm="giac")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^2*(c*cos(f*x + e))^m, x)