3.911 \(\int \cos ^m(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=157 \[ -\frac{A \sin (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) \cos ^{m+2}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right )}{d (m+n+2) \sqrt{\sin ^2(c+d x)}} \]

[Out]

-((A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]
^2]*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])) - (B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^n*Hypergeom
etric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])

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Rubi [A]  time = 0.0842588, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {20, 2748, 2643} \[ -\frac{A \sin (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) \cos ^{m+2}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right )}{d (m+n+2) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]

[Out]

-((A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]
^2]*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])) - (B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^n*Hypergeom
etric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

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

Mathematica [A]  time = 0.241586, size = 130, normalized size = 0.83 \[ -\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \left (A (m+n+2) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )+B (m+n+1) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right )\right )}{d (m+n+1) (m+n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]

[Out]

-((Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(2 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (
3 + m + n)/2, Cos[c + d*x]^2] + B*(1 + m + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2
, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m + n)*(2 + m + n)))

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Maple [F]  time = 1.993, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+B\cos \left ( dx+c \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^m*(b*cos(d*x+c))^n*(A+B*cos(d*x+c)),x)

[Out]

int(cos(d*x+c)^m*(b*cos(d*x+c))^n*(A+B*cos(d*x+c)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^n*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c))^n*cos(d*x + c)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^n*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*cos(d*x + c) + A)*(b*cos(d*x + c))^n*cos(d*x + c)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**m*(b*cos(d*x+c))**n*(A+B*cos(d*x+c)),x)

[Out]

Integral((b*cos(c + d*x))**n*(A + B*cos(c + d*x))*cos(c + d*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^m*(b*cos(d*x+c))^n*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c))^n*cos(d*x + c)^m, x)