3.57 \(\int (b \cos ^m(c+d x))^n \, dx\)

Optimal. Leaf size=78 \[ -\frac{\sin (c+d x) \cos (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1) \sqrt{\sin ^2(c+d x)}} \]

[Out]

-((Cos[c + d*x]*(b*Cos[c + d*x]^m)^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]))

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Rubi [A]  time = 0.0347181, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3208, 2643} \[ -\frac{\sin (c+d x) \cos (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x]^m)^n,x]

[Out]

-((Cos[c + d*x]*(b*Cos[c + d*x]^m)^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]))

Rule 3208

Int[(u_.)*((b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sin[e + f*x
])^n)^FracPart[p])/(c*Sin[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Sin[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

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 \left (b \cos ^m(c+d x)\right )^n \, dx &=\left (\cos ^{-m n}(c+d x) \left (b \cos ^m(c+d x)\right )^n\right ) \int \cos ^{m n}(c+d x) \, dx\\ &=-\frac{\cos (c+d x) \left (b \cos ^m(c+d x)\right )^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)}}\\ \end{align*}

Mathematica [A]  time = 0.061062, size = 72, normalized size = 0.92 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x]^m)^n,x]

[Out]

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

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Maple [F]  time = 0.274, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{m} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*cos(d*x+c)^m)^n,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 )^{m}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cos(d*x + c)^m)^n, 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 )^{m}\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cos(d*x + c)^m)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*cos(c + d*x)**m)**n, 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 )^{m}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cos(d*x + c)^m)^n, x)