3.209 \(\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt{\sin ^2(c+d x)}} \]

[Out]

(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]
^2]*Sin[c + d*x])/(d*(5 + 3*m)*Sqrt[Sin[c + d*x]^2])

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Rubi [A]  time = 0.0278641, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {20, 2643} \[ -\frac{3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3),x]

[Out]

(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]
^2]*Sin[c + d*x])/(d*(5 + 3*m)*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 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))^{2/3} \, dx &=\frac{(b \cos (c+d x))^{2/3} \int \cos ^{\frac{2}{3}+m}(c+d x) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}\\ &=-\frac{3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5+3 m);\frac{1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.117181, size = 82, normalized size = 1. \[ -\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (m+\frac{5}{3}\right );\frac{1}{2} \left (m+\frac{11}{3}\right );\cos ^2(c+d x)\right )}{d \left (m+\frac{5}{3}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3),x]

[Out]

-((Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*Hypergeometric2F1[1/2, (5/3 + m)/2, (11/3 + m)/2,
Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(d*(5/3 + m)))

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}\, 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))^(2/3),x)

[Out]

int(cos(d*x+c)^m*(b*cos(d*x+c))^(2/3),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 )\right )^{\frac{2}{3}} \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))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c))^(2/3)*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 )\right )^{\frac{2}{3}} \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))^(2/3),x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c))^(2/3)*cos(d*x + c)^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(cos(d*x+c)**m*(b*cos(d*x+c))**(2/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \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))^(2/3),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^(2/3)*cos(d*x + c)^m, x)